BIOGRAPHICAL MEMOIRS

Christopher Charles Heyde 1939–2008

This memoir was originally published in Historical Records of Australian Science, vol.20, no.1, 2009.

Introduction

An account of the Heyde family is followed by a description of Chris’s childhood, schooling and university training at Sydney and the ANU. Chris spent most of his academic career at the ANU, CSIRO and Columbia University. He made an outstanding contribution to probability theory and its applications. His theoretical work focused mainly on the laws of large numbers, branching processes, martingale theory, estimation theory and, most recently, financial mathematics. He also had a lasting interest in the history of probability and statistics. Chris received considerable recognition, including Fellowship of the Australian Academy of Science and of the Academy of Social Sciences in Australia, as well as Membership of the Order of Australia.

Some Family History

Chris Heyde was born in Sydney, Australia on 20 April 1939, the son of Gilbert Christoph von der Heyde and Alice Danne Wessing. He died in Canberra on 6 March 2008, from the effects of metastatic melanoma.

Chris’s Father’s Family—the Heyde Family

Chris’s paternal great-great-grandfather, Jacob Christoph von der Heyde (1792– 1839) was a master mason and his great-grandfather, Wilhelm von der Heyde, was born in the Lüneburger Heide area of Hanover, Germany, in 1826. He arrived in Adelaide in 1848 and moved through Melbourne, Hobart and Sydney, prospering as a tobacco merchant. He built a handsome house at Strathfield, a Sydney suburb, and was elected mayor of the municipality.

Chris’s grandfather Charles, Wilhelm’s eldest son, was born in Manly in 1879 and went to school in Germany for two years. He became managing director of a big combine of tobacco companies (British Australasian Tobacco) and was there for the whole of his working life. He married Gertrude (Truda) Philip, who was born in Armidale, New South Wales, of Scottish stock.

Chris’s father, Gilbert Christoph von der Heyde (1914–2000), was born in Sydney, the fourth child of his parents, and was also known as Chris. He was an adventurous skier as a young man. His father offered to send him to university at age 18, at the time of the great Depression, but he turned down the offer in order to start in the tobacco industry as a junior clerk, remaining in the industry for nineteen years. During this time he studied in the evenings for the Sydney Technical College’s Diploma of Chemical Engineering, as well as taking a range of science subjects out of interest. He soon became recognized as a very good manager and joined the newly established Australian Institute of Management (AIM), serving on AIM panels for twenty years and being elected a Fellow (FAIM). In 1951, he became head of Unilever’s management consulting department in Australia, staying on till 1970.

In 1966 Chris senior developed Modapts, MODular Arrangement of Predetermined Time Systems. This could be used to model different ways of doing a task and gave ‘fair times’ for work that could be used by both management and unions. It was applied in a range of jobs and organizations in various countries. It was so successful that he left Unilever to continue work on it full-time, with revised and enlarged editions being produced.

Chris senior was innovative from the start of his career. His essential curiosity about how things worked led him to explore phenomena well before they became widely known to the public, including early computers. His most animated stories were about discovering the causes of malfunction, non-function or dissatisfaction. The work on family history from which some of these notes are extracted was a manifestation of his general curiosity about history. An obituary by his son from his second marriage, Victor, appeared in the Sydney Morning Herald (V. Heyde, 2000).

Chris’s Mother’s Family—the Wessing Family

Chris’s maternal great-grandfather, Peter Wheisel (Hveisel) Wessing, a master baker, was born in Grenaa, Denmark. He married Rasmine Olavia Schjodt, who was born in 1838 in Ebeltoft, Denmark, the daughter of a ship’s captain. They emigrated to Tasmania in 1872. Their expectation was to become farmers, though they had received a good education—it is said that they had experience in law and medicine. Rasmine had a reputation for nursing/medical skills in her community near Hobart. Four of their children travelled with them.

Chris’s grandfather, Carl Sophus Wessing (1871–1949) was still a baby when his parents departed for Australia, while he remained with his grandmother in Denmark. He joined his family in Hobart at the age of thirteen years. He married Nella Marie Fredericca Neilson (Nelsson), who was of Swedish descent on her father’s side and whose mother had sung in the Stockholm Opera House. They had five daughters and a son before Nella died, when the eldest daughter Charlotte (Lottie, who was fifteen years older than Chris’s mother Alice) took on the mothering role. Carl worked in mines in Queenstown as a powder monkey, setting explosives. He observed the quality of the ore and used to telegraph Lottie with instructions about buying shares in mining companies. He eventually earned enough to buy a significant family property at Summerhill Road in Hobart, where he established a plant nursery and produced grafted fruit trees for the then flourishing orchard industry in Tasmania. At one stage he also owned two houses at Battery Point.

Carl and Nella Wessing with their Scandinavian background hosted the Norwegian explorer RoaldAmundsen in Hobart in early 1912, when Amundsen in the vessel Fram was on his way back from Antarctica after becoming the first person to reach the South Pole in December 1911.

Chris’s mother, Alice Danne Wessing (1908–1975), was the fifth daughter in the family and had one younger brother. Her sisters all worked at home duties in their father’s home before marriage, but Alice wanted to work outside the home. She qualified in typing and shorthand, winning two gold medals for speed, and became a court reporter in Hobart. She had two overseas voyages to Europe and was an adventurous skier, as was her younger brother Charles, both in Tasmania and on the mainland at Mount Kosciuszko. She met her future husband Gilbert Christoph von der Heyde on the skiing fields, and they married in 1936. After they separated, and once her son Chris was established in school, she undertook secretarial duties with a legal practitioner in Sydney. Chris spent many happy times in Tasmania during his childhood and youth with his mother’s family.

Chris’s Formation: Childhood to High School

When Chris was born his parents were living at Cambridge Avenue, Vaucluse, in Sydney. During 1942 the family moved to Cheltenham, where Chris lived until he moved to Canberra as a PhD student in 1962.

Chris’s parents separated when he was four or five years old. His parents both recalled that one day in their garden Chris made a critical comment to them, questioning why the relationship was not harmonious: his father referred to this incident as being in no small way a factor in his decision to leave. Chris’s mother closed the door on the relationship as far as Chris was concerned, and provided him with a very untroubled and secure upbringing.

For most of his schooling Chris attended Barker College, Hornsby (a Sydney suburb). Through the primary years he was often ill and did not do particularly well in his studies, in large part owing to his having missed large amounts of work.

As Chris became older he began to forge relations through St John’s Anglican church in Beecroft; he became active in the youth fellowship and the choir (in which he sang as a tenor), and attended church moderately often. Whatever was on in the sporting domain took top priority, and his mother supported his sporting interests, never questioning him about them.

He played rugby at school when he was 13–15, in the B team (out of A, B, C). He played one game in the A team because they needed his speed, but he had too great a sense of self-preservation and preferred to grab the opponents rather than ‘ankle-tap’ them as the fearless and highly prized tacklers do.

For a few years Chris swam competitively, his best results coming in Under 13 and Under 14 years competitions. He was best for his age at Barker then, but by the Under 15 stage Chris could see that others were putting in less work and getting better results. He decided it was then ‘a mug’s game’ to persist and gave up competitive swimming. Harry Hay was Chris’s favourite swimming coach; his other coaches were more impersonal and the squads were larger. Chris won the Harry Hay trophy, which was inaugurated as a memorial to Harry. This was a handicap race at the Spit, in which swimmers were rated, and Chris performed best in relation to the rating he was given. He held the huge cup for a year. This was the sporting trophy that meant most to him. Chris performed all the swimming strokes competitively but was best at freestyle. He often raced against Murray Rose, whom no-one ever beat and who went on to win several gold medals in the Olympics in 1956 and 1960. Chris used to come around 6th place in the State championships.

Chris made a smooth transition to athletics although he had started these late, at about 13, swimming having been his preferred activity until then. He went on to be best for age in Under 14, Under 15 and Under 16 at Barker. He had quite a reputation for courage in running relays. He ran last in his team and always caught up with some of the competition, running with such passion that people recognized his effort. At the Combined Schools Association meets, he was second in the 200 m and 800 m. Chris started with the Ryde/Hornsby athletics club at around the age of 14. Here his athletics career ended at age 15 when he tore his left Achilles tendon, probably because he had not warmed up enough before doing some long jumping. After this injury, he had significant difficulty in walking for a while and it was clear that there would be no more athletics in the near future.

At school, he was suddenly sedentary and began to take an interest in his lessons. Fortunately it was not too late. A few teachers were more inspirational than others: the good teachers tried to teach the subject rather than the curriculum, and were dedicated to their discipline. Among them was Robert Finlay who taught English. Chris was in the honours English stream and enjoyed it, gaining a good foundation in writing. The other inspirational teacher was Gordon Miller, who taught mathematics and science. He noticed that Chris was able to do things others could not; Chris’s scores moved up from about 65% to more than 95% in tests. When it came to the Leaving Certificate, Chris took the examinations twice. The first time was with no subjects at the honours level and he scored As in English, Mathematics I and II and Physics, and Bs in French and one other subject, a respectable score. The second time, he trained for the honours level, taking a completely different curriculum for these— Mathematics I Hons, Mathematics II Hons, Physics Hons, English, which was compulsory, and Applied Mathematics in which he was almost self-taught. He gained high results in Mathematics I and II, not quite so high in Physics, and As for English and Applied Mathematics. He came comfortably in the top 100 in the state, though there was no consequent entitlement in the way of scholarships or bursaries. However, he won a Commonwealth Scholarship, for which he had also qualified the previous year. A chief advantage in doing the Leaving Certificate for the second time was that he achieved a stronger position for first-year courses at university.

Chris was active in the school cadet corps, becoming an Underofficer in the final year of school. This was a quasi-commissioned rank with a paper mandate from the Army, essentially a platoon leader. Chris enjoyed Cadets increasingly as he became more senior, even to the extent of wondering about going to Duntroon Military College and joining the Army. His Uncle Phillip Heyde was part of the inspiration. But this was before Chris became a high achiever in Mathematics, when it was obvious to him that Science was the course to study.

Chris was Dux of Barker College in 1956. In his final year, he was also a prefect and coach/manager of the swimming team. He was generally liked at school. Though he did not have many close friends there— no-one shared his academic interests—he did not feel isolated.

University Studies

Chris entered the Faculty of Science at the University of Sydney at the beginning of 1957 on his Commonwealth Undergraduate Scholarship; in those days in Australia these scholarships exempted their holders from university fees and sometimes provided a living allowance. For students doing Science, they covered the three years of a pass-level Bachelor’s degree and Fourth-Year Honours, which was then essentially training for research.

Chris went into the Mathematics Honours stream (doing Pure and Applied Mathematics combined) and Physics Honours. He was also in a class of 500 starting Chemistry from scratch, while Geology was his fourth subject, chosen because he preferred studying rocks to cutting up frogs. He could see that geology provided a general explanation of how the Earth fits together and found it very interesting. Chris’s attitude to geology wasn’t altogether appreciated by his tutors: after a field trip in the Blue Mountains, Chris’s report received the boldly written comment, ‘Too much cogitation, too little observation!’ He received good Credits in Geology and Chemistry and a Distinction in Mathematics, the latter being a somewhat disappointing result for him. He was in a very high-powered Mathematics class—everyone in it had been in the top 100 in the Leaving Certificate examinations. People took this course even if they were going to do Medicine. After first year, Chris gained High Distinctions in all his Mathematics subjects, but he was clearly one of the top students even in the first-year class, and he remembered competing intellectually with other students, which contrasted with his time at Barker.

The Chemistry and Mathematics Departments at the University of Sydney were held in high esteem at the time, both having Heads who were FRSs. Physics was expanding rapidly under its new Head, Harry Messel, an entrepreneur who managed to have six new Professors appointed and also obtained SILLIAC, among the first computers in Australia. Mathematics was in the Physics building, and Chris worked in the Physics library. When Harry Messel came through, students left because he always had a strongly-smelling cigar clamped between his teeth. By second year, Chris had given up on Physics, his enthusiasm having been severely dented when staff responded to questions along the lines, ‘you don’t need to know that’. In contrast, the quality of modelling in the Applied Mathematics Department, headed by Professor Keith Bullen, a seismologist, was extremely high. The approach was very rational, looking at what features one wanted to capture and what were the key elements needed for doing this, what could be left out, what had to be in—always aiming for the simplest possible model but all logically founded. Chris felt the intellectual rigour and found it very satisfying; he had never heard anyone teach modelling the way Bullen did and he later tried to do something similar in his own teaching and practice. He stressed what a model needed to capture, what it was supposed to do and what one hoped to get out of it, what were the inescapable building blocks. His own finance model, for which he later obtained a patent, is couched in these terms.

Bullen and Professor Thomas G. Room, Head of Pure Mathematics, were both Cambridge-trained and both FRS’s. However they were at loggerheads and did not speak to each other. Through third year, Chris did not enjoy Pure Mathematics although he scored well in it, but he was enjoying Mathematical Statistics with Harry Mulhall, as he also had in his second year. Mulhall was a very good teacher, quite systematic, and although he was not research-orientated he was aware of current research. Until Professor H. O. Lancaster was appointed at the end of Chris’s third year and a Department of Statistics was set up, Mulhall was the only person teaching Statistics at the University of Sydney. With Lancaster’s appointment, Fourth-Year Honours in Statistics became an option, and Chris agonized between Honours in Statistics and Honours in Applied Mathematics. He decided on Statistics and was a member of the first Honours class, together with Murray Aitkin, Mohammed (David) Hamdan, Reg Armson, and Brother Bennett (Seneta 2002; Seneta and Eagle-son 2004). In contrast to Harry Mulhall, Oliver Lancaster proved a somewhat less gifted teacher—except for a few students. Lancaster talked about his mathematical research and showed through his reasoning where he was going and what he wanted to achieve. Most undergraduate students could not adapt to this, but Chris understood and ‘ate it up’; it was hugely helpful to him as research training.

Chris was a meticulous keeper of records and files, and his lecture notes from his undergraduate years at Sydney have survived. They reflect the quality of undergraduate education there at the time and are historically interesting as an indication of Chris’s gestation as an eminent statistician. Undergraduate, presumably second-year reference books were P. G. Hoel’s Introduction to Mathematical Statistics (2nd edn.); A. Mood’s Introduction to the Theory of Statistics; H. Cramér’s Elements of Probability Theory and W. Feller’s Introduction to Probability Theory and Its Applications, Volume 1. For Mathematical Statistics III (in Fourth-Year Honours) there are the following sets of lecture notes: Rupert Leslie’s course: ‘Order Statistics’; Eve Bofinger’s course: ‘Analysis of Variance and Design of Experiments’; Ian Stewart’s course: ‘Quality Control and Sampling Theory’; Oliver Lancaster’s courses: ‘Analysis of Variance’ and ‘Distribution of Quadratic Forms’; and Harry Mulhall’s courses: ‘Distribution Theory’ and ‘Inference’. There is also a Pure Mathematics IIIA junior paper by Chris entitled ‘Asymptotics’, surely a predictor of the shape of things to come.

Chris had no significant social life at university as his days were filled with a packed lecture schedule, laboratory work every afternoon, and travel to and from home. For recreation he was active in the Hornsby Rifle Club on Saturday afternoons for about two years.

Chris gained his BSc with first-class honours in Mathematical Statistics and the University Medal from the University of Sydney in 1960, the degree being conferred in 1961. In 1958 and 1960, he also won the prizes for statistics of the New South Wales Branch of the Statistical Society of Australia.

Oliver Lancaster wanted Chris to stay on in his department and a scholarship for the Master’s degree was arranged. In those days, the MSc was a natural next step in research training after Fourth-Year Honours. Chris continued his studies at Sydney and received his MSc in 1962 for a thesis on the ‘Theory of characteristic functions and the classical moment problem’. His note from this period (3), in which he showed that the lognormal distribution is not determined by its moments, became a classic, eventually receiving the ultimate accolade of being mentioned in the Bible of advanced probability theory, the book of Feller (1971), p. 227. When his Master’s thesis was virtually complete, Chris visited Canberra with Oliver Lancaster and met P. A. P. (Pat) Moran and other members of Moran’s Department of Statistics at the Institute of Advanced Studies of the Australian National University (the ANU). In 1961, Chris had won a Commonwealth Postgraduate Research Scholarship and he took this with him when he moved to Moran’s department in early 1962. Moran’s department admirably fulfilled the key purpose for which the ANU had been created, namely to allow talented Australian students to pursue their doctoral research in Australia rather than having to go overseas: it attracted the cream of Australian students in Statistics for several decades (Gani 2005).

Chris’s PhD thesis was entitled ‘Results related to first passage time problems and some of their applications’. His nominal supervisor was J. E. (Jo) Moyal, but Chris worked mainly on his own, following some suggestions made by Pat Moran. He was awarded his PhD in Statistics in 1965. Eventually Chris was to write Pat Moran’s biographical memoir (139).

Chris met his wife-to-be, Thelma Elizabeth (Beth) James, at University House, the residence for unmarried PhD students at the ANU, in 1963, when they were both postgraduate students. Beth’s first memory of Chris was when he played the role of Roman centurion in Bernard Shaw’s play ‘Androcles and the Lion’, performed as a Sunday night play-reading in the basement of the Eastern Annex of University House. He wore laced-up Roman sandals, carried a garbage bin lid for a shield, and had a dustpan brush strapped to the top of his head. His role was not large, but the audience gave him a rousing reception.

Beth had won the Lilley Medal (first place in the Queensland Scholarship Examination, which pleased her teacher parents) and later studied Science at the University of Queensland, winning the University Medal. She won a General-Motors-Holden’s postgraduate scholarship to undertake a PhD in Biochemistry at the ANU in the John Curtin School of Medical Research, and moved to Canberra in 1963. She shared with Chris a love of nature and being out in the bush that she had gained during her childhood in various parts of Queensland, while Chris had done so from growing up on the edge of what is now Lane Cove National Park in Sydney. They also shared a commitment as Christians in the Anglican tradition.

Chris led bushwalks for the residents of University House, and caused some concern to his friends on one occasion when he was out reconnoitring for such an outing, because he did not return on schedule. He had underestimated the time required to get back and had to spend a night in the bush, but was able to walk out by himself when daylight returned. He went skiing when he could, and made an igloo with friends on one occasion. He, Bob Pidgeon and Peter Brockwell ran a sluice box on Araluen Creek, hoping to pick up some alluvial gold from the reef that had never been found. They did collect tiny bits of gold, but made the mistake of trying an old prospector’s method for turning these into a ‘nugget’ by mixing them with mercury and putting the mixture inside a potato that was then cooked; the theory was that the gold collected in a lump in the centre and the mercury diffused, but in practice it all diffused and they ended up with only a very poisonous potato.

Beth and Chris became engaged in May 1964. It was decided that Beth would finish her thesis in Canberra by August 1965, and that they would marry in Brisbane in September 1965. In August 1964 Chris submitted his PhD thesis and sailed from Sydney for the USA, where he joined Joe Gani, who had also been a member of Pat Moran’s department, and Uma Prabhu, then of the University of Western Australia, who had both moved to the Department of Statistics at Michigan State University, East Lansing. The three of them attempted to build up teaching and research in stochastic processes. When Joe Gani left towards the end of 1965 to take up the Chair of Probability and Statistics at the University of Sheffield in the UK, Chris followed him there as a Lecturer.

University of Sheffield 1965–1968; Australian National University 1968–1975

Chris returned to Australia in time for the wedding with Beth in September 1965 at St Colomb’s Anglican Church, Clayfield, Queensland. Their honeymoon began with a week spent at Heron Island and continued on board the P&O vessel Orsova, sailing from Sydney to Southhampton en route to Sheffield.

On arriving in Sheffield with their Volkswagen ‘Beetle’, which had travelled with them, both Chris and Beth settled in to work, Beth on a research grant in the laboratory of one of the Biochemistry staff, Stanley Ainsworth. At weekends they explored the countryside in England and Scotland.

Three months in the summer of 1966 were spent in Denmark. Here Chris worked at Aarhus University in the department of Ole Barndorff-Nielsen, with whose later work, especially as it related to financial mathematics, Chris was to become familiar. Touring around Denmark at the weekends, Chris and Beth included a visit to Grenaa on the north-east coast of Jutland where Chris’s maternal grandfather had been born. From Denmark, Chris and Beth travelled to Moscow in August 1966 for Chris to attend the 1966 International Mathematical Congress.

The first of their two children, Neil, was born on 12 June 1967, just before they moved to Manchester. Chris had been promoted to Special Lecturer in charge of the Statistical Laboratory at the University of Manchester from September 1967, when the Manchester-Sheffield School of Probability and Statistics was formed. Statistics at Manchester had come under Joe Gani’s aegis following the departure to Cambridge of Professor Peter Whittle.

Chris was offered three positions in Australia and decided to take up one at the ANU, a Readership in Ted Hannan’s Department of Statistics in the School of General Studies (SGS). The family returned to Australia in September 1968. Chris had by then produced some thirty papers, a dominant theme of which was the refinement of classical limit theory involving large and small deviations, rates of convergence and domains of attraction, while displaying a breadth of interest in the contemporary issues in probability. There were strong links between Hannan’s teaching department and Pat Moran’s purely research department in the Institute of Advanced Studies (IAS). The SGS department, which also had considerable strength in research, stimulated Chris’s interests in new directions, notably the theory of branching processes, statistical inference for them, and population genetics models related to them. He published several papers jointly authored with Eugene Seneta on these topics and others with Ted Hannan on time series analysis, in addition to a number of papers that he wrote alone. In this work, a principal focus of Chris’s was the martingale concept. He was to become widely known for his work on the theory and application of martingale methods, not least in estimation for stochastic processes. In 1973, he was awarded a DSc by the ANU ‘after due examination of his published work in the field of mathematical statistics and probability’.

Chris’s period in the SGS department also saw the genesis of his interest in the history of probability and statistics in company with Eugene Seneta. Both were influenced by Bienaymé’s 1845 discovery of the criticality theorem of branching processes, to which they had been led by remarks of Oliver Lancaster. Their book on Bienaymé was effectively a history of probability and statistics in the nineteenth century, perhaps the first of a modern resurgence of books on the history of statistics.

The period at the ANU saw the beginnings of intense editorial activity on Chris’s part, both at home and internationally. This is described in Gani and Seneta (2008).

Soon after returning to Canberra, Beth found part-time work at the John Curtin School of Medical Research (JCSMR), in the Biochemistry Department in which she had done her PhD. The first Moon landing on 20 July 1969 was being broadcast on television in the week or so before their second child, Eric, was born. Beth returned to part-time work at JCSMR in January 1970.

Chris arranged to take sabbatical leave at Stanford University for a year beginning with the northern Fall semester in 1972. In 1973, after attending St John’s Anglican Church at Reid, with which both Chris and Beth were familiar from student days, since 1968, they joined the Anglican community that was then meeting in the Aranda school hall near their new home, a link that continued at the time of Chris’s death.

CSIRO Division of Mathematics and Statistics 1975–1983; University of Melbourne 1983–1986

In January 1975 Chris joined the CSIRO Division of Mathematics and Statistics, of which Joe Gani had just become Chief. Meanwhile Beth was able to go back to full-time work, taking up a research fellowship at JCSMR. Chris was at first a Senior Principal Research Scientist and, from 1977, Chief Research Scientist and Assistant Chief of the Division. He took over as Acting Chief in 1981, when Joe Gani left the Division.

Chris was elected a Fellow of the Australian Academy of Science in 1977. His proposer was Pat Moran, with Ted Hannan as seconder. The citation submitted a few years earlier read:

Dr Heyde is an internationally recognized authority on the classical theory of probability. His principal contributions are concerned with the problems of convergence to normality, laws of large numbers and martingale theory. He has also worked on renewal theory, queueing theory and stochastic models for chemical processes. In the last 10 years he has published a large body of work which shows great originality and technical power.

At the time, nine new ordinary Fellows were being elected each year; Chris was elected in the face of very intense competition.

In September 1983 Chris became Professor and Chairman of the Department of Statistics at the University of Melbourne. He proved to be an excellent Chairman who strongly encouraged the pursuit of research and the use of computer facilities by staff and students. Instrumental in creating the Statistical Consulting Centre, he gave strong support to its director and staff. In 1985, he succeeded in obtaining a very large grant from the Australian Government to support a Key Centre for Statistical Science, a joint enterprise of LaTrobe, Monash and Melbourne Universities and the Royal Melbourne Institute of Technology (RMIT); Chris became founding director of this Centre.

Figure 1. Chris Heyde, President of the Statistical Society of
Australia, presents the Pitman Medal for 1980 to Oliver Lancaster.
From left to right: Bill Kruskal, Oliver Lancaster, Chris Heyde,
Edwin Pitman.

During this period he took on new editorial responsibilities. These included: Associate Editor of the International Statistical Review, 1980–1987; Joint Editor of The Mathematical Scientist, 1982–1984, and Associate Editor from 1984; Coordinating Editor of Advances in Applied Probability and the Journal of Applied Probability, 1983–1989 and Editor-in-Chief of these journals, 1990–2008 (jointly with Soren Asmussen from May 2005). He was one of the editors of the Australian Mathematical Society Lecture Series from 1984, and one of the editors of the Springer Monograph Series in Probability and its Applications from 1985. In 1984, following the death of Norma McArthur, one of the initial trustees of the Applied Probability Trust (APT), Chris was appointed as one of the four APT Trustees. His counsel was always balanced and wise and will be sorely missed. Further detail may be found in Gani and Seneta (2008) and Nash (2008).

Other commitments for the period included: member of the organizing committee for Section 8, Mathematical Sciences, 46th ANZAAS Conference, Canberra, January 1975; organizer of the 8th International Conference for Stochastic Processes and their Applications, Canberra, July 1978; member of the Committee for Conferences on Stochastic Processes, 1973–1983, and Chairman 1979–1981 and 1981–1983 (two terms); Alternative Director of SIROMATH Pty Ltd, October 1980– July 1981, and Director, August 1981– January 1983; member of the scientific advisory committee for the Australian Government Inquiry into the Possible Effects of Herbicides on Vietnam Veterans and their Families, 1980–1984; member of the Science and Industry Forum of the Australian Academy of Science from 1980; chairman of the Australian Statistics Policy Committee, 1980–1984; member of the Queen Elizabeth II Fellowships Committee, 1983; and member of the Australian Subcommission of the International Commission for Mathematical Instruction, 1984–1987.

Figure 2. From left to right: Chris Heyde, Ted Hannan, Joe Gani and
Eugene Seneta. ANU, Canberra, 6 January 1994.

Return to the ANU 1986–2008; Columbia University 1993–2008

In May 1986, Chris returned to the ANU to become Head of the Department of Statistics in the Research School of Social Sciences of the Institute of Advanced Studies, serving from July 1986 to December 1988. Pat Moran had retired in 1982 and Ted Hannan, Head until July, retired in December 1986.

The two Mathematics Departments (SGS and IAS), the IAS Department of Statistics and the ANU’s Special Research Centre for Mathematical Analysis soon afterwards underwent an important structural change in which Chris played a pivotal role, coming together to form the ANU School of Mathematical Sciences (since renamed the Mathematical Sciences Institute). This was the third attempt at bringing the ANU mathematicians together, but while the ANU was happy to hold the School up as a shining example of cooperation between its research and undergraduate teaching arms, the Institute and the Faculties, it did not provide it with adequate support. This contrasted, Chris noted, with the approach to new enterprises that he later found at Columbia University, where strong backing was given to the burgeoning area of financial mathematics. Chris was the Foundation Dean of the new School, serving from January 1989 to January 1992. From February 1992 to January 2005, he was Professor of Statistics in the School (later Institute), as a member of its Stochastic Analysis Group.

From 1993, he was also a professor in the Department of Statistics at Columbia University, New York. He taught there for their Fall semester each year (September to December) until 2007, and was the director of the Columbia Center for Applied Probability. He was intensely active in this role. To commemorate his contribution to the University, Columbia held an ‘Applied Probability Day in Honor of Chris C. Heyde’ on Saturday 28 June 2008. He was appointed Professor Emeritus of Statistics on 6 March 2008, the University President stating that ‘this reaffirmation of his importance to our scholarly community only begins to recognize his extraordinary contributions to Columbia’. More information on his contributions at Columbia may be found in Glasserman and Kou (2006).

On the occasion of Chris’s 65th birthday, a conference in his honour (CMA National Research Symposium on Probability Theory and its Applications, 22–23 April 2004) was held at the ANU, followed by a dinner in the Great Hall of University House. At this time, his colleagues, friends and former students offered him a Festschrift (Gani and Seneta 2004) as a token of the deep esteem and affection in which he was held by the mathematical and statistical communities in Australia and elsewhere.

In the midst of his very full academic life, Chris found time for travel, relaxation and recreation with his family. There were adventure tours exploring remote and beautiful regions of Australia; relaxation, such as cruising in Scandinavia; and frequent weekend and vacation retreats at South Durras on the New South Wales coast. He also very much enjoyed the many opportunities for overseas visits linked with his international responsibilities and research contributions.

During this part of his life, Chris continued to take a serious interest in the development of Mathematics and Statistics, both in Australia and internationally. He was chairman of the Executive Committee of the Australian Foundation for Science in 1990– 1992, and was a director of the Foundation, 1992–1999. A member of the council of the Australian Mathematical Society, 1980– 1983, he became its Vice-President in 1981. He was Vice-President of the International Statistical Institute (ISI, to whose membership he had been elected in 1972) in 1985– 1987 and again in 1993–1995; a member of the ISI’s Bernoulli Society Council in 1979– 1987, its President-elect in 1983–1985 and its President in 1985–1987.

Chris was a council member of the Canberra Branch of the Statistical Society of Australia (SSA), 1973–1983, and Branch President, 1987–1989. He put much effort into the job of Director of the National Mathematical Sciences Congress in 1988. This was held in Canberra, under the auspices of the Australian Bicentennial Authority. When at the University of Melbourne, he was a member of the Victorian Branch Council in 1984–1986 and Branch President in 1985–1986. He was a member of the SSA’s Central Council, 1973–1986, and the Society’s Federal President in 1985– 1986. He was a member of the Australian Mathematics Competition Board in 1981– 1992 and of the Board of its successor, the Australian Mathematics Trust, from 1992. His publications list contains invited articles that attest to his on-going concern about the public perception and future of mathematical and statistical science, presented from his authoritatively perceptive standpoint.

Chris served the Australian Academy of Science in a variety of ways. He was a member of Sectional Committee 1 (Mathematics), 1978–1982 (chairman, 1980– 1982), and also a member of Council in 1986–1993, Vice-President in 1988–1989, and Treasurer in 1989–1993.

In 1994, Chris was awarded the Hannan Medal of the Australian Academy of Science, and in 1995 the Thomas Ranken Lyle Medal. This period of his life brought other well-deserved rewards: the Pitman Medal of the Statistical Society of Australia; a DSc (honoris causa) of his alma mater, the University of Sydney; membership of the Order of Australia; the Centenary Medal of the Australian Government, and election as a Fellow of the Academy of Social Sciences in Australia.

In a remarkable presentation for the 19th Pfizer Colloquium, Chris was filmed for the American Statistical Association’s Distinguished Statisticians Archive (14), following in the footsteps of earlier eminent probabilists and statisticians. His talk encompassed the manifold areas of his experience in the service of statistics and probability. A highlight was his recommendation for proper supervision and mentoring of graduate students. Chris candidly expressed his views about the statistical profession, its growth over its golden decades (1950–1980), its current state and its likely future. This included topics such as a decreasing and ageing membership in statistical associations, the decline of ‘Mathematical Statistics’ as a discipline and of Departments of Statistics as separate entities, and the increasingly important roles in the practice of statistics and applied probability played by disciplines such as bioinformatics, data mining, and the mathematical treatment of financial risk.

Glasserman and Kou (2006) contains an excellent published conversation with Chris about his professional career, valuable in particular for the description of his perceptions and the evolution of his scientific thinking.

Research

Chris’s research covered a huge variety of topics, testifying to a great breadth of interest and a remarkable ability to assimilate new directions in probability. His publications include works on the moment problem, first passage problems, random walks, the iterated logarithm law, recurrent events, enzyme reactions, queueing theory, branching processes, martingale theory, estimation theory particularly for branching and stochastic processes, genetic balance and gene survival, invariance principles, weak convergence of probability measures, the Hawkins random sieve, reproduction rates and clutch sizes of birds, outbreaks of rare infections, random trees and stemma construction in philology, long-range dependence, fractals and random fields, random matrices in demographic projections, quasi-likelihood methods, estimation for queueing processes and processes with long-range dependence, inference for time series, robustness of limit theorems, risk assessment for catastrophic events, fractal scaling and generalizations of the Black-Scholes model in financial mathematics.

One of the central themes of Chris’s research in his first post-PhD period at the ANU was the probabilistic concept of a martingale, which derives from a gambling context whence the name comes. A sequence of random variables {Xn}, n= 0, 1, 2,... is said to be a martingale if E(Xn+1 | Xn,Xn-1-1,..., X0) = Xn. (That is, if the expected value of a random variable at time n + 1 given information on the entire past is the actual value observed at time n.) A martingale difference sequence is then {Yn}, n = 1,2,... where Yn = Xn - Xn-1.

Towards the end of his life, Chris listed what he considered his five favourite papers. These were:

  1. The paper (3) on the moments of the lognormal distribution that we have already mentioned above.
  2. A joint paper with Ted Hannan (45), in which it is shown that the best linear predictor is the best predictor if the innovations are martingale differences. This was one of the very early papers that made it clear that martingales would play an important role in statistics. (A martingale difference sequence is a generalization that allows for dependence, of the classical statistical context of independent zero-mean random variables.)
  3. An invited paper (47) expounding the emerging role that martingales were to play in probability. In particular, this contains a Central Limit Theorem for martingales. (A martingale can be regarded as a sum of martingale differences, and hence as a generalization of a sum of independent random variables. The classical Central Limit theory is framed in terms of a limiting normal Gaussian distribution for a normed sum of independent random variables.)
  4. A joint paper with Y. Yang (167), clarifying the concept of long-range dependence. This concept was of special interest to Chris in the last prevailing direction of his research, which involved modelling the probabilistic behaviour of financial assets.
  5. The paper (176) that introduced the fractal activity time geometric Brownian motion (FATBGM) risky asset model. This was the starting point for what is now a very large body of work by Chris, his students and his colleagues (for example (180), (183) and (200)), on models that capture subtle aspects of empirically observed financial asset data sequences.

This list is, however, an excessively modest account of Chris’s achievements. To it one might readily add the following contributions:

  1. Limit theorems in branching processes, as in a joint paper with E. Seneta (41).
  2. Rates of convergence in the Central Limit Theorem, as considered in (65).
  3. Inference in stochastic processes, as studied in (80).
  4. The pioneering text (2), Martingale Limit Theory and its Application, written with P. G. Hall.
  5. The clear exposition (8), as a book, of quasi-likelihood and its application.

We must also include Chris’s persistent interest in history as exemplified by his two books, I. J. Bienaymé: Statistical Theory Anticipated written with E. Seneta (1), and Statisticians of the Centuries edited with E. Seneta (11). This listing gives an overview of the immense span of his interests.

(15) includes commentaries on various areas of his research.

A most recent sphere of Chris’s activity was financial modelling. During the 1990s, while at Columbia University, his mathematical focus swung towards the stochastic modelling of long-range dependence and its effect on the observed behaviour of risky assets such as stocks. His ideas on dependence in models for financial returns, and the treatment of the heavy-tailedness of their distribution, have been hugely influential.

Chris’s key paper from this period is undoubtedly (176), followed by his paper with S. Liu (180). A more recent publication with N. N. Leonenko (202) is destined to become a classic. All build on his firm and long-held belief that the generalized symmetric t-distribution, because of its power-law tails, is the correct distribution for modelling stock market returns. With the support of Columbia University, he applied for and was granted a US Patent (13) arising out of this work.

Chris’s graduate MSc/MPhil students at the ANU by research thesis included P. G. Hall (later FAA FRS), R. J. Adler, I. M. Johnstone, C. W. Lloyd-Smith, I. S. McRae, A. M. Currie and A. Sly.

He supervised many PhDs, some of them jointly: V. Rohatgi at Michigan State University; P. D. Feigin, D. B. Pollard, R. Maller, D. J. Scott, J. R. Leslie, R. Gay, Y.-X. Lin, W. Dai, J. M. Senyonyi-Mubiru, B. Colbert, S. Hurst, S. M. Tam and B. Wong at the ANU; and Yanmei Yang and Olivier Nimeskern at Columbia. Supervision of Ross Maller (now himself a professor at the ANU) at SGS was continued by Eugene Seneta when Chris left the ANU for CSIRO. Ross now supervises Chris’s continuing students.

Epilogue

Chris was diagnosed with hairy-cell leukaemia eleven years before his death and underwent periods of treatment, including participation in a clinical trial in the USA, followed by periods of blessed remission. He completed his normal activities at Columbia University in the Fall of 2007, but early in 2008, in Canberra, metastatic melanoma was diagnosed. In an email message dated 20 January 2008 to one of us, he wrote: ‘Whatever happens, I certainly feel that I have had a fortunate life. I will be happy to have more,…but if not, I have had a good innings and can go in peace.’

We both saw him a few days before a scheduled hip replacement operation to relieve pain from pathological fracture. He died in Canberra just over a day after the operation, in the early morning of 6 March 2008.

The funeral was held at Holy Covenant Anglican Church, Dexter Street, Cook, ACT, not far from the Heyde home at Aranda, on Thursday 13 March, in the presence of family and many mourners and friends. His ashes lie in the grounds of Norwood Park Crematorium in Canberra, marked by a memorial plaque with an infinity sign and a butterfly.

Acknowledgements

Our thanks are due to Dr Beth Heyde who provided detailed family history. She also transcribed speaking notes and gave us access to Chris Heyde’s meticulously stored archival material. We also thank Rosanne Walker, Librarian, Australian Academy of Science.

References

  1. Feller,W.(1971)An Introduction to Probability Theory and Its Applications, Vol. 2. (Wiley: New York).
  2. Gani, J. (2005) ‘Fifty Years of Statistics at the Australian National University, 1952– 2002’, Historical Records of Australian Science, 16(1), 31–44.
  3. Gani, J. (1994) ‘Edward James Hannan, 1921– 1994’, Historical Records of Australian Science, 10(2), 173–185.
  4. Gani, J. and Seneta, E. (eds) (2004) Stochastic Methods and Their Applications: Papers in Honour of Chris Heyde (J.Appl. Prob., Special Vol. 41A) [Introduction by editors: pp. vii–x].
  5. Gani, J. and Seneta, E. (2008) ‘Obituary: Christopher Charles Heyde, AM, DSc, FAA, FASSA’, Journal of Applied Probability, 45, 587–592.
  6. Glasserman, P. and Kou, S. (2006) ‘A conversation with Chris Heyde’, Statistical Science, 21(2), 286–298.
  7. Heyde, V. (2000) ‘Chris Heyde: Expert on work, coin collector, 1914–2000’, Sydney Morning Herald, 29 November 2000.
  8. Nash, L. (2008) ‘Chris Heyde: An Appreciation’, Journal of Applied Probability, 45, 593–594.
  9. Seneta, E. (2002) ‘In Memoriam: Emeritus Professor Henry Oliver Lancaster AO FAA, 1 February 1913–2 December 2001’, Australian and New Zealand Journal of Statistics, 44(4), 385–400.
  10. Seneta, E. and Eagleson, G. K. (2004) ‘Henry Oliver Lancaster, 1913–2001’, Historical Records of Australian Science, 15(2), 223–250.

Bibliography

Books, Patent and Film

  1. I. J. Bienaymé: Statistical Theory Anticipated (with E. Seneta). Springer-Verlag, New York, 1977. xiv + 172 pp.
  2. Martingale Limit Theory and its Application (with P. G. Hall). Academic Press, New York, 1980. xii + 308 pp.
  3. Studies in Modelling and Statistical Science: Papers in Honour of J. Gani. C. C. Heyde (ed.). Austral. J. Statist., Special Volume 30A, 1988. ix + 309 pp.
  4. Bicentennial History Issue. C. C. Heyde and E. Seneta (eds). Austral. J. Statist., Special Volume 30B, 1988. ix + 130 pp.
  5. Youth Employment and Unemployment. W. Dunsmuir, C. C. Heyde and I. McRae (eds). Austral. J. Statist. Special Volume 31B, 1989. iii + 225 pp.
  6. Branching Processes: Proceedings of the First World Congress. C. C. Heyde (ed.). Springer Lecture Notes in Statistics 99, 1995. vi + 179 pp.
  7. Athens Conference on Applied Probability and Time SeriesAnalysis. Volume 1:Applied Probability. C. C. Heyde, Yu. V. Prohorov, R. Pyke and S. T. Rachev (eds). Springer Lecture Notes in Statistics 114, 1996. x + 448 pp.
  8. Quasi-Likelihood and Its Application: General Theory of Optimal Parameter Estimation. Springer-Verlag, NewYork, 1997. ix + 235 pp.
  9. Probability Towards 2000. L. Accardi and C. C. Heyde (eds). Springer Lecture Notes in Statistics 128, 1998. xi + 356 pp.
  10. Special Issue on Long-Range Dependence. V. V. Anh and C. C. Heyde (eds). Austral. J. Statist, 80, 1999. vi + 290 pp.
  11. Statisticians of the Centuries. C. C. Heyde and E. Seneta (eds). Springer-Verlag, New York, 2001. xii + 500 pp.
  12. Selected Proceedings of the Symposium on Inference for Stochastic Processes. I. V. Basawa, C. C. Heyde and R. L. Taylor (eds). IMS Lecture Notes–Monographs Series, Institute of Mathematical Statistics, Beach-wood, Ohio, Volume 37, 2001. 356 pp.
  13. US Patent 6643631: ‘Method and system for modeling financial markets and assets using fractal activity time’, 4 November 2003.
  14. ‘A Futuristic View on a Half-Century of Statistics and Applied Probability’: The 19th Pfizer Colloquium, presented at the University of Connecticut-Storrs, filmed November 4, 2005. ‘Filming of Distinguished Statisticians’ series, American Statistical Association [DVD deposited in Basser Library, Australian Academy of Science, Canberra].
  15. Selected Works of C. C. Heyde. R. Maller (ed.). Selected Works in Probability and Statistics. Springer-Verlag, New York. [To appear in 2010.]

Papers and Articles

1963

  1. Some remarks on the moment problem I, Quarterly J. Math. (2nd series) Oxford, 14, 91–96.
  2. Some remarks on the moment problem II, Quarterly J. Math. (2nd series) Oxford, 14, 97–105.
  3. On a property of the lognormal distribution, J. Roy. Statist. Soc. B, 25, 392–393.

1964

  1. Two probability theorems and their applications to some first passage problems, J. Austral. Math. Soc., 4, 214–222.
  2. On the stationary waiting time distribution in the queue G1/G/1, J. Applied Prob., 1, 173–176.

1966

  1. Some results on small deviation probability convergence rates for sums of independent random variables, Canadian J. Math., 18, 656–665.
  2. Some renewal theorems with applications to a first passage problem, Ann. Math. Statist., 37, 699–710.

1967

  1. A pair of complementary theorems on convergence rates in the law of large numbers (with V. K. Rohatgi), Proc. Camb. Phil. Soc., 63, 73–82.
  2. Asymptotic renewal results for natural generalization of classical renewal theory, J. Roy. Statist. Soc. B, 29, 141–150.
  3. Some local limit results in fluctuation theory, J. Austral. Math. Soc., 7, 455–464.
  4. A limit theorem for random walks with drift,J. Applied Prob., 4, 144–150.
  5. A contribution to the theory of large deviations for sums of independent random variables, Z. Wahrscheinlichkeitstheorie, 7, 303–308.
  6. On the influence of moments on the rate of convergence to the normal distribution, Z. Wahrscheinlichkeitstheorie, 8, 12–18.
  7. On large deviation problems for sums of random variables which are not attracted to the normal law, Ann. Math. Statist., 38, 1575–1578.

1968

  1. A further generalization of the arc-sine law, J. Austral. Math. Soc., 8, 369–372.
  2. On almost sure convergence for sums of independent random variables, Sankhya Ser.A, 30, 353–358.
  3. Variations on a renewal theorem of Smith, Ann. Math. Statist., 39, 155–158.
  4. On large deviation probabilities in the case of attraction to a non-normal stable law, Sankhya Ser. A, 30, 253–258.
  5. On the converse to the iterated logarithm law, J. Applied Prob., 5, 210–215.
  6. An extension of the Hájek-Rényi inequality for the case without moment conditions, J. Applied Prob., 5, 481–483.
  7. On the growth of a random walk, Ann. Inst. Stat. Math., 20, 315–321.

1969

  1. On extremal factorization and recurrent events, J. Roy. Statist. Soc. B, 31, 72–79.
  2. A derivation of the ballot theorem from the Spitzer-Pollaczek identity, Proc. Camb. Phil. Soc., 65, 755–757.
  3. Some properties of metrics in a study on convergence to normality, Z. Wahrscheinlichkeitstheorie, 11, 181–192.
  4. On a fluctuation theorem for processes with independent increments II, Ann. Math. Statist., 40, 688–691.
  5. On the maximum of sums of random variables and the supremum functional for a stable process, J. Applied Prob., 6, 419–429.
  6. A note concerning behaviour of iterated logarithm type, Proc. Amer. Math. Soc., 23, 85–90.
  7. On extended rate of convergence results for the invariance principle, Ann. Math. Statist., 40, 2178–2179.
  8. A stochastic approach to a one substrate one product enzyme reaction in the initial velocity phase (with Elizabeth Heyde), J. Theor. Biol., 25, 159–172.

1970

  1. On some mixing sequences in queueing theory, Operations Research, 18, 312–315.
  2. Extensions of a result of Seneta for the super-critical Galton-Watson process, Ann. Math. Statist., 41, 739–742.
  3. Characterization of the normal law by the symmetry of a certain conditional distribution, Sankhya Ser. A, 32, 115–118.
  4. On the implication of a certain rate of convergence to normality, Z. Wahrscheinlichkeitstheorie, 16, 151–156.
  5. A rate of convergence result for the super-critical Galton-Walton process, J. Applied Prob., 7, 451–454.
  6. On the departure from normality of a certain class of martingales (with B. M. Brown), Ann. Math. Statist., 41, 2161–2165.

1971

  1. On the growth of the maximum queue length in a stable queue, Operations Research, 19, 447–452.
  2. Stochastic fluctuations in a one substrate one product enzyme system: are they ever relevant? (with Elizabeth Heyde), J. Theor. Biol., 30, 395–404.
  3. Some central limit analogues for super-critical Galton-Walton processes, J. Applied Prob., 8, 52–59.
  4. An invariance principle and some convergence rate results for branching processes (with B. M. Brown), Z. Wahrscheinlichkeitstheorie, 20, 271–278.
  5. Some almost sure convergence theorems for branching processes, Z. Wahrscheinlichkeitstheorie, 20, 189–192.
  6. Analogues of classical limit theorems for the super-critical Galton-Walton process with immigration (with E. Seneta), Math. Biosci., 11, 249–259.
  7. Improved classical limit analogues for Galton-Walton processes with or without immigration (with J. R. Leslie), Bull. Austral. Math. Soc., 5, 145–155.

1972

  1. On the influence of moments on approximations of portion of a Chebyshev series in central limit convergence (with J. R. Leslie), Z. Wahrscheinlichkeitstheorie, 21, 255–268.
  2. Estimation theory for growth and immigration rates in a multiplicative process (with E. Seneta), J. Applied Prob., 9, 235–256.
  3. On limit theorems for quadratic functions of discrete time series (with E. J. Hannan), Ann. Math. Statist., 43, 2058–2066.
  4. The simple branching process, a turning point test and a fundamental inequality: a historical note on I. J. Bienaymé (with E. Seneta), Biometrika, 59, 680–683.
  5. Martingales: a case for a place in the statistician’s repertoire. Invited Paper, Austral. J. Statist., 14, 1–9.

1973

  1. An iterated logarithm result for martingales and its application in estimation theory for autoregressive processes, J. Applied Prob., 10, 146–157.
  2. On the uniform metric in the context of convergence to normality, Z.Wahrscheinlichkeitstheorie, 25, 83–95.
  3. Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments (with D. J. Scott), Annals of Probability, 1, 428–436.
  4. Revisits for transient random walk, Stoch. Proc. Appl., 1, 33–51.

1974

  1. An iterated logarithm result for autocorrelations of a stationary linear process, Annals of Probability, 2, 328–332.
  2. Notes on estimation theory for growth and immigration rates in a multiplicative process (with E. Seneta), J. Applied Prob., 11, 572–577.
  3. Limit theory for stationary processes via approximating martingales, Abstract of invited paper given to the 3rd Conference on Stochastic Processes and their Applications, Sheffield, August 1973, Adv. Applied Prob., 6, 196–197.
  4. On estimating variance of the offspring distribution in a simple branching process, Adv. Applied Prob., 6, 421–433.
  5. On the central limit theorem for stationary processes, Z. Wahrscheinlichkeitstheorie, 30, 315–320.
  6. On martingale limit theory and strong convergence results for stochastic approximation procedures, Stoch. Proc. Appl., 2, 359–370.

1975

  1. A supplement to the strong law of large numbers, J. Applied Prob., 12, 173–175.
  2. Remarks on efficiency in estimation for branching processes, Biometrika, 62, 49–55.
  3. Kurtosis and departure from normality, in Statistical Distributions in Scientic Work,Vol. 1, Models and Structures, Eds G. P. Patil, S. Kotz and J. K. Ord, D. Riedel Publ. Co., Dordrecht, 193–201.
  4. On efficiency and exponential families in stochastic process estimation (with P. D. Feigin) in Statistical Distributions in Scientifjc Work, Vol. 1, Models and Structures, Eds G. P. Patil, S. Kotz and J. K. Ord, D. Riedel Publ. Co., Dordrecht, 1975, 227–240.
  5. On the central limit theorem and iterated logarithm law for stationary processes, Bull. Austral. Math. Soc., 12, 1–8.
  6. The genetic balance between random sampling and random population size (with E. Seneta), J. Math. Biol., 1, 317–320.
  7. Martingale methods in estimation theory, Abstract of invited paper given to the 4th Conference on Stochastic Processes and their Applications, Toronto, August 1974, Adv. Applied Prob., 7, 235–237.
  8. A non-uniform bound on convergence to normality, Annals of Probability, 3, 903–907.
  9. Bienaymé (with E. Seneta), Bull. Int. Statist. Inst., 46, Book 2, 318–331.

1976

  1. Estimation of parameters for stochastic processes, Abstract of invited paper given to the First Conference of the CSIRO Division of Mathematics and Statistics, Sydney, February 1975, The Mathematical Scientist, Supplement 1, 3–5.
  2. On a unified approach to the law of the iterated logarithm for martingales (with P. G. Hall), Bull. Austral. Math. Soc., 14, 435–447.
  3. Asymptotic properties of maximum likelihood estimators for stochastic processes (with I. V. Basawa and P. D. Feigin), Sankhya, Ser. A, 38, 259–270.
  4. On asymptotic behaviour for the Hawkins random sieve, Proc. Amer. Math. Soc., 56, 277–280.
  5. On moment measures of departure from the normal and exponential laws (with J. R. Leslie), Stoch. Proc. Appl., 4, 317–328.

1977

  1. The effect of selection on genetic balance when the population size is varying, Theoret. Population Biol., 11, 249–251.
  2. On the rate of convergence in the martingale convergence theorem,Abstract of invited paper given to the 6th Conference on Stochastic Processes and their Applications, Tel Aviv, June 1976, Adv. Applied Prob., 9, 196.
  3. Some rate of convergence results for the martingale convergence theorem, Abstract of invited paper given to the 3rd Conference of the Statistical Society of Australia, Melbourne, August 1976, The Mathematical Scientist, 2, 141–143.
  4. An optimal property of maximum likelihood with application to branching process estimation, Bull. Int. Statist. Inst., 47, Book 2, 407–417.
  5. On central limit and iterated logarithm supplements to the martingale convergence theorem, J. Applied Prob., 14, 758–775.

1978

  1. Bienaymé, lrenée Jules (with E. Seneta), Dictionary of Scientific Biography, 15, 30–33.
  2. Uniform bounding of probability generating functions and the evolution of reproduction rates in birds (with H.-J. Schuh), J. Applied Prob., 15, 243–250.
  3. A log log improvement to the Riemann hypothesis for the Hawkins random sieve, Annals of Probability, 6, 870–875.
  4. On an optimal asymptotic property of the maximum likelihood estimator of a parameter from a stochastic process, Stoch. Proc. Appl., 8, 1–9.
  5. On an explanation for the characteristic clutch size of some bird species, Adv. Applied Prob., 10, 723–725.

1979

  1. Applications of stochastic processes: some general principles and their illustration, The Mathematical Scientist, 4, 1–8.
  2. On asymptotic posterior normality for stochastic processes (with I. M. Johnstone), J. Roy. Statist. Soc. B, 41, 184–189.
  3. On central limit and iterated logarithm results for subadditive processes, Abstract of paper given to the 8th Conference on Stochastic Processes and their Applications, Canberra, July 1978, Adv. Applied Prob., 11, 283–284.
  4. On assessing the potential severity of an outbreak of a rare infectious disease: a Bayesian approach, Austral. J. Statist., 21, 282–292.

1980

  1. On a probabilistic analogue of the Fibonacci sequence, J. Applied Prob., 17, 1079–1082. Abstract in Adv. Applied Prob., 12, 282.

1981

  1. Rates of convergence in the martingale central limit theorem (with P. Hall), Annals of Probability, 9, 395–404.
  2. On Fibonacci (or lagged Bienaymé-Galton-Watson) branching processes, J. Applied Prob., 18, 583–591.
  3. Invariance principles in statistics, International Statistical Review, 49, 143–152.
  4. Looking forward into the 1980s: a personal view of the problems and prospects for the statistical profession. Presidential Address to the Statistical Society of Australia, August 1980. Austral. J. Statist., 23, 1–14.
  5. On the survival of gene represented in a founder population, J. Math. Biol., 12, 91–99.
  6. Trends in the statistical sciences. The Belz Lecture for 1980. Austral. J. Statist., 23, 273–286.

1982

  1. The Australian Journal of Statistics, in Encyclopedia of Statistical Sciences, Eds N. L. Johnson and S. Kotz. Wiley, New York, Vol. 1, 147–148.
  2. Estimation in the presence of a threshold theorem: principles and their illustration for the traffic intensity, in Statistics and Probability: Essays in Honour of C. R. Rao, Eds G. Kallianpur, P. R. Krishnaiah and J. K. Ghosh. North-Holland, Amsterdam, 317–323.
  3. The effect of differential reproductive rates on the survival of a gene represented in a founder population, in Essays in Statistical Science, Eds J. Gani and E. J. Hannan, J. Applied Prob. Special Vol. 19A, 19–25.
  4. Optimal estimation of the criticality parameter of a supercritical branching process having random environments (with A. G. Pakes), J. Applied Prob., 19, 415–420.
  5. On the asymptotic behaviour of random walks on a anisotropic lattice, J. Statist. Phys., 27, 721–730.
  6. Statistics (with J. Gani), in Mathematical Sciences in Australia 1981, Australian Academy of Science, Canberra, 60–70.
  7. On the number of terminal vertices in certain random trees with application of stemma construction in philology (with D. Najock), J. Applied Prob., 19, 675–680.
  8. The asymptotic behaviour of a random walk on a dual medium lattice (with M. Westcott and E. R. Williams), J. Statist. Phys., 28, 375–380.
  9. Further results on the survival of a gene represented in a founder population (with D. J. Daley and P. G. Hall), J. Math. Biol., 14, 355–363.

1983

  1. Invariance principles and functional limit theorems, in Encyclopedia of Statistical Sciences, Eds N. L. Johnson and S. Kotz. Wiley, New York, Vol. 4, 225–228.
  2. Law of the Iterated Logarithm, in Encyclopedia of Statistical Sciences, Eds N. L. Johnson and S. Kotz. Wiley, New York, Vol. 4, 528–530.
  3. Law of Large Numbers, in Encyclopedia of Statistical Sciences, Eds N. L. Johnson and S. Kotz. Wiley, New York, Vol. 4, 566–568.
  4. Limit Theorem, Central, in Encyclopedia of Statistical Sciences, Eds N. L. Johnson and S. Kotz. Wiley, New York, Vol. 4, 651–655.
  5. The sociology of discovery in pre-20th century probability and statistics. 1. Period pre1830, The Mathematical Scientist, 8, 1–10.
  6. The effect on humans of exposure to herbicides: a contentious contemporary problem in statistical inference, The Mathematical Scientist, 8, 63–73.
  7. An alternative approach to asymptotic results on genetic composition when the population size is varying, J. Math. Biol., 18, 163–168.

1984

  1. On limit theorems for gene survival, in Limit Theorems in Probability and Statistics. Ed. P. Révész, Colloquia Mathematica János Bolyai, 36, Vol. II, North-Holland, Amsterdam, 573–586.
  2. On the asymptotic equivalence of L_p metrics for convergence to normality (with T. Nakata), Z.Wahrscheinhchkeitstheorie, 68, 97–106.

1985

  1. On some new probabilistic developments of significance to statistics: martingales, long range dependence, fractals and random fields, in A Celebration of Statistics. The ISI Centenary Volume. Eds A. C. Atkinson and S. E. Fienberg, Springer, NewYork, 355–368.
  2. Multidimensional and central limit theorems, in Encyclopedia of Statistical Sciences, Eds N. L. Johnson and S. Kotz. Wiley, New York, Vol. 5, 643–646.
  3. Confidence intervals for demographic projections based on products of random matrices (with J. E. Cohen), Theoret. Population Biol., 27, 120–153.
  4. On inference for demographic projection of small populations, in Proceedings of the Berkeley Conference in Honour of Jerzy Neyman and Jack Kiefer, Eds L. Le Cam and R. A. Olshen, Wadsworth, Monterey, Calif. Vol. 1, 215–223.
  5. On macroscopic stochastic modelling of systems subject to criticality, The Mathematical Scientist, 10, 3–8.
  6. An asymptotic representation for products of random matrices, Stoch. Proc. Appl., 20, 307–314.

1986

  1. Probability Theory (Outline), in Encyclopedia of Statistical Sciences, Eds N. L. Johnson and S. Kotz. Wiley, New York, Vol. 7, 248–252.
  2. Quantile transformation methods, in Encyclopedia of Statistical Sciences, Eds N. L. Johnson and S. Kotz. Wiley, New York, Vol. 7, 432–433.
  3. On the use of time series representations of population models, in Essays in Time Series and Allied Processes. J. Applied Prob., Special Vol. 23A, 345–353.
  4. Random matrices, in Encyclopedia of Statistical Sciences, Eds N. L. Johnson and S. Kotz. Wiley, New York, Vol. 7, 549–551.
  5. Random sum distributions, in Encyclopedia of Statistical Sciences, Eds N. L. Johnson and S. Kotz. Wiley, New York, Vol. 7, 565–567.
  6. Optimality in estimation for stochastic processes under both fixed and large sample conditions, in Probability Theory and Mathematical Statistics. Proceedings of the Fourth Vilnius Conference, Eds V. Yu, Prohorov, V. A. Statulevicius, V. V. Sazonov and B. Griegelionis. VNU Sciences Press, Utrecht, Vol. I, 535–541.
  7. Comment on the paper “Applications of Poisson’s work” by I. J. Good, Statistical Science, 1, 176–177.

1987

  1. On combining quasi-likelihood estimating functions, Stoch. Proc. Appl., 25, 281–287.
  2. Quasi-likelihood and optimal estimation (with V. P. Godambe), Int. Statist. Rev., 55, 231–244.
  3. Optimal robust estimation for discrete time stochastic processes (with P. M. Kulkarni), Stoch. Proc. Appl., 26, 267–276.

1988

  1. Some thoughts on stationary processes and linear time series analysis, A Celebration of Applied Probability. J. Applied Prob., 25A, 309–318.
  2. Asymptotic efficiency results for the method of moments with application to estimation for queueing processes, in Queueing Theory and its Application. Liber Amicorum for J. W. Cohen, Eds O. J. Boxma and R. Syski, CWI Monograph No. 7, North-Holland, Amsterdam, 405–412.
  3. Fixed sample and asymptotic optimality for classes of estimating functions, Contemporary Mathematics (Amer. Math. Soc.), 80, 241–247.
  4. Official statistics in the late colonial period leading on to the work of the first Commonwealth Statistician, G. H. Knibbs, Austral. J. Statist., 30B, 23–43.

1989

  1. On asymptotic quasi-likelihood estimation (with R. Gay), Stoch. Proc. Appl., 31, 223–236.
  2. Comments in the discussion of the paper “An extension of quasi-likelihood estimation” by V. P. Godambe and M. E.Thompson, Austral. J. Statist, 22, 160.
  3. On efficiency for quasi-likelihood and composite quasi-likelihood methods, in Statistical Data Analysis and Inference, Ed. Y. Dodge, North-Holland, Amsterdam, 209–213.
  4. Quasi-likelihood and optimality for estimating functions: some current unifying themes, The Fisher Lecture, 1989, Bull. Int. Statist. Inst., 53, Book 1, 19–29.

1990

  1. On a class of random field models which allows long-range dependence (with R. Gay), Biometrika, 77, 401–403.
  2. Estimating population size from multiple recapture experiments (with N. Becker), Stoch. Proc. Applic., 36, 77–84.
  3. A view from the mathematical sciences, in Proceedings of the Forum on the Profile of Australian Science, ASTEC, 41–44.

1991

  1. Approximate confidence zones in an estimating function context (with Y.-X. Lin), in Estimating Functions, Ed. V. P. Godambe, Oxford University Press, Oxford, 161–168.

1992

  1. Patrick Alfred Pierce Moran1917–1988. Biographical Memoirs of Fellows of the Royal Society, 37, 366–379, and Historical Records of Australian Science, 9, 17–30.
  2. “Thoughts on the modelling and identification of random processes and fields subject to possible long-range dependence” (with R. Gay), in Probability Theory, Eds L. H.Y. Chen, K. P. Choi, K. Hu and J.-H. Lou, de Gruyter, Berlin, 75–81.
  3. The promotion and development of applied probability: a note on the contributions of J. Gani, in Selected Proceedings of the Symposium on Applied Probability, Eds I. V. Basawa and R. L. Taylor, IMS Monograph Series 18, Hayward, Calif., 9–11.
  4. New developments in inference for temporal stochastic processes, Austral. J. Statist, 33, 121–129.
  5. Some results on inference for stationary processes and queueing systems, in Queueing and Related Models, Eds U. N. Bhat and I. V. Basawa, Oxford University Press, Oxford, 337–345.
  6. On best asymptotic confidence intervals for parameters of stochastic processes, Ann. Statist., 20, 603–607.
  7. On quasi-likelihood methods and estimation for branching processes and hetero-scedastic regression models (with Y.-X. Lin), Austral. J. Statist., 34, 199–206.

1993

  1. Weak convergence of probability measures, in Soviet Encyclopedia of Mathematics, Kluwer, Dordrecht, Vol. 9, 448–449.
  2. Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence (with R. Gay), Stoch. Proc. Applic., 45, 169–182.
  3. Asymptotics for two-dimensional anisotropic random walks, in Stochastic Processes. A Festschrift in Honour of Gopinath Kallianpur, Eds S. Cambanis, J. K. Ghosh, R. L. Karandikar and P. K. Sen, Springer, New York, 125–130.
  4. Quasi-likelihood and general theory of inference for stochastic processes, in 7th International Summer School on Probability Theory and Mathematical Statistics, Lecture Notes, Eds A. Obretenov and V. T. Stefanov, Science Culture Technology Publishing, Singapore, 122–152.
  5. Allan, Frances Elizabeth (1905–1952), Australian Dictionary of Biography, Vol. 13, 26–27.
  6. Optimal estimating functions and Wedderburn’s quasi-likelihood (with Y.-X. Lin), Comm. Statist.Theory Meth., 22, 2341–2350.
  7. Societies, membership and the future of ISI, Bull. Int. Statist. Inst., 49, Book 1, 79–84.
  8. On constrained quasi-likelihood estimation (with R. Morton), Biometrika, 80, 755–761.

1994

  1. On the effects of noise in systems involving products of random matrices, in Recent Advances in Statistics and Probability, Eds M. L. Puri and J. P. Vilaplana, VSP International Science Publishers, Netherlands, 277–284.
  2. A quasi-likelihood approach to estimating parameters in diffusion type processes, in Studies in Applied Probability, Eds J. Galambos and J. Gani, J. Applied Prob., 31A, 283–290.
  3. A quasi-likelihood approach to the REML estimating equations, Statistics and Probability Letters, 21, 381–384.
  4. Edward James Hannan, 29 January 1921– 7 January 1994, Austral. Math. Soc. Gazette, 21, 121–122.

1995

  1. On asymptotic optimality of estimating functions (with K. Chen), Austral. J. Statist, 48, 101–112.
  2. A conversation with Joe Gani, Statistical Science, 10, 214–230.
  3. On the robustness of limit theorems (with W. Dai), Bull. Internat. Statist. Inst., Vol. 56, Book 2, 549–555.

1996

  1. Quasi-likelihood and generalizing the E-M algorithm (with R. Morton), J. Roy. Statist. Soc. B, 58, 317–327.
  2. On the robustness to small trends of estimation based on the smoothed periodogram (with W. Dai), J. Time Series Analysis, 17, 141–150.
  3. Ito’s formula with respect to fractional Brownian motion and its application (with W. Dai), J. Applied Math. Stochastic Analysis, 9, 439–448.
  4. On the use of quasi-likelihood for estimation in hidden-Markov random fields, Austral. J. Statist, 50, 373–378.

1997

  1. On spaces of estimating functions (with Y.-X. Lin), Austral. J. Statist, 63, 255–264.
  2. Avoiding the likelihood, in Selected Proceedings of the Symposium on Estimating Equations, Eds I. V. Basawa, V. P. Godambe and R. L. Taylor, IMS Lecture Notes–Monograph Series, Vol. 32, 35–42.
  3. On defining long-range dependence (with Y. Yang), J. Applied Prob., 34, 939–944.
  4. Asymptotic optimality, in Encyclopedia of Mathematics, Suppl. No. 1, Ed. M. Hazewinkel, Kluwer, Dordrecht, 69–70.

1998

  1. The chasm between the scientist and the media, in Statistics, Science and Public Policy, Eds A. M. Herzberg and I. Krupka, Proceedings of a Conference held at Herstmonceaux Castle, Hailsham, UK, April 10–13, 1996, Queen’s University, Kingston, Ontario, Canada, Chapter 5, 31–34.
  2. Role of national academies, in Statistics, Science and Public Policy, Eds A. M. Herzberg and I. Krupka, Proceedings of a Conference held at Herstmonceaux Castle, Hailsham, UK, April 10–13, 1996, Queen’s University, Kingston, Ontario, Canada, Chapter 11, 65–68.
  3. Multiple roots in general estimating equations (with R. Morton), Biometrika, 85, 954–959.
  4. Fate more than roll of the dice, The Canberra Times, August 6, p. 14.
  5. Risk assessment for catastrophic events, in Statistics, Science and Public Policy II, Hazards and Risks. Proceedings of a Conference held at Queen’s University, Kingston, Ontario, Canada, April 23– 25, 1997, Queens’s University, Kingston, Ontario, Canada, Chapter 13, 85–87.

1999

  1. Prediction via estimating functions (with A. Thavaneswaran), Austral. J. Statist, 77, 89–101.
  2. Stochastic models for fractal processes (with V. V. Anh and Q. Tieng), Austral. J. Statist, 80, 123–135.
  3. A risky asset model with strong dependence through fractal activity time, J.Applied Prob., 36, 1234–1239.

2000

  1. Comment on the paper “Eliminating multiple root problems in estimation” by C. G. Small, J. Wang and Z. Yang, Statistical Science, 15, 334–335.

2001

  1. A note on filtering for long memory processes (with A. Thavaneswaran), in Stable Models in Finance and Econometrics, Math. Comput. Modelling, 34, 1139–1144.
  2. The changing scene for higher education in science: a view from the statistical profession, in Statistics, Science and Public Policy V, Society, Science and Education. Proceedings of a Conference held at Herstmonceaux Castle, Hailsham, UK, April 26–29, 2000, Queens’s University, Kingston, Ontario, Canada, Chapter 2, 13–23.
  3. Empirical realities for a minimal description risky asset model. The need for fractal features (with S. Liu), Principal Invited Paper, Mathematics in the New Millenium Conference,Yonsei University, Seoul, Korea, October 2000, J. Korean Math. Soc., 38, 1047–1059.
  4. An overview of the Symposium on Inference for Stochastic Processes, in Selected Proceedings of the Symposium on Inference for Stochastic Processes, Eds I. V. Basawa, C. C. Heyde and R. L. Taylor, IMS Lecture Notes–Monograph Series, Vol. 37, 3–7.
  5. Shifting paradigms in inference, in Selected Proceedings of the Symposium on Inference for Stochastic Processes, Eds I. V. Basawa, C. C. Heyde and R. L. Taylor, IMS Lecture Notes–Monograph Series, Vol. 37, 9–21.
  6. Fractal scaling and Black-Scholes: the full story. A new view of long-range dependence in stock prices (with S. Liu and R. Gay), JASSA. Journal of the Australian Society of Security Analysts, Issue 1, 29–32.
  7. John Graunt, in Statisticians of the Centuries, Eds C. C. Heyde and E. Seneta, Springer-Verlag, New York, 14–16.
  8. George Handley Knibbs, in Statisticians of the Centuries, Eds C. C. Heyde and E. Seneta, Springer-Verlag, New York, 257–260.
  9. Agner Krarup Erlang, in Statisticians of the Centuries, Eds C. C. Heyde and E. Seneta, Springer-Verlag, New York, 328–330.
  10. Parameter estimation of stochastic processes with long-range dependence and intermittency (with J. T. Gao, V. V. Anh and Q. Tieng), J. Time Series Anal., 22, 517–535.
  11. On option pricing with the FATGBM risky asset model, Bull. Int. Statist. Inst., 53rd Session, Contributed Papers, 59, Book 3, 313–314.

2002

  1. Probabilistic models, in Encyclopedia of Environmetrics, Eds A. H. El-Shaarawi and W. W. Piegorsch, Wiley, Chichester, Vol. 3, 1637–1638.
  2. Statistical estimation of nonstationary Gaussian processes with long-range dependence and intermittency (with J. T. Gao and V. V. Anh), Stoch. Proc. Appl., 99, 295–321.
  3. Obituary: Richard L. Tweedie, Appl. Stoch. Models Bus. Ind., 18, 1–2.
  4. Dynamic models of long-memory processes driven by Lévy noise (with V. V. Anh and N. N. Leonenko), J. Applied Prob., 39, 730–747.
  5. On modes of long-range dependence, J. Applied Prob., 39, 882–888.
  6. Challenges facing statistics for the 21st century, in Advances in Statistics, Combinatorics and Related Areas, Eds C. Gulati, Y.-X. Lin, S. Mishra and J. Rayner, World Scientific, Singapore, 238–244.

2003

  1. Some efficiency comparisons for estimators from quasi-likelihood and generalized estimating equations (with S. Liu), Mathematical Statistics and Applications: Festschrift in Honour of Constance van Eeden, Eds M. Moore, S. Froda and C. Leger, IMS Lecture Notes–Monographs Series, Vol. 42, 357–374.

2004

  1. Comments on the paper “Evidence functions and the optimality of the law of likelihood” by S. Lele, in The Nature of Scientific Evidence: Empirical, Statistical and Philosophical Considerations, Eds M. L. Taper and S. Lele, University of Chicago Press, Chicago, 203–205.
  2. Asymptotics and criticality for a correlated Bernoulli process, Aust. N. Z. J. Stat., 46, 53–57.
  3. The central limit theorem. Encyclopedia of Actuarial Science. (Eds-in-Chief J. Teugels and B. Sundt.) Wiley, Chichester, 1, 255–260.
  4. On the controversy over tailweight of distributions (with S. G. Kou), Oper. Res. Letters, 32, 399–408.
  5. On the martingale property of stochastic exponentials (with B. Wong), J. Applied Prob., 41, 654–664.
  6. On subordinated market models (with A. Irle), in Proceedings of the International Sri Lankan Statistical Conference: Visions of Futuristic Methodologies, December Eds B. M. de Silva and N. Mukhopadhyay, Postgraduate Inst. Science, Univ. Peradenya, Sri Lanka, 1–15.

2005

  1. Student processes (with N. N. Leonenko), Adv. Applied Prob., 37, 342–365.

2006

  1. On the problem of discriminating between the tails of distributions (with K. Au), in Contributions to Probability and Statistics: Applications and Challenges, Proceedings of the University of Canberra International Statistical Workshop, April 2005, Eds P. Brown, S. Liu and D. Sharma, World Scientific, Singapore, 246–258.
  2. On changes of measure in stochastic volatility models (with B. Wong), Journal of Applied Mathematics and Stochastic Analysis, 18130.

2008

  1. Non-standard limit theorem for infinite variance functionals (with A. Sly), Annals of Probability, 36, 796–805.
  2. On estimation in conditionally heteroskedastic time series models under non-normal distributions (with S. Liu), Statistical Papers, 49, 455–469.
  3. A cautionary note on model choice and the Kullback–Leibler information (with K. Au), J. Statist. Theor. Practice, 2, 221–232.

2009

  1. Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims (with D. Wang), Adv. Applied Prob., 41, 206–224.

Forthcoming

  1. A cautionary note on modeling with fractional Lévy flights (with A. Sly), Physica A: Statistical Mechanics and Its Applications. [To appear]
  2. What is a good external risk measure: Bridging the gaps between robustness, subadditivity and insurance risk measures (with S. G. Kou and X. H. Peng). [Submitted]

E. Seneta, School of Mathematics and Statistics, FO7, University of Sydney, NSW 2006. Corresponding author. Email: eseneta@maths.usyd.edu.au

J. M. Gani, Mathematical Sciences Institute, Australian National University, Canberra ACT 0200.