**John Kingman**'s paternal grandfather was a coal miner in Somerset. He was determined that his sons would have a good education and not need to become miners so John's father Frank E T Kingman was able to study at the University of Bristol, where he was awarded his Ph.D. He moved to London on taking up a scientific post with the government and there married John's mother Maud Elsie Harley who came from a London family. They had two children, and John and his brother were brought up on the outskirts of London.

John was educated at Christ's College, Finchley in London where he won a scholarship to Cambridge. Of his school days he said [1]:-

I had several very good teachers at school, and received a lot of encouragement in my ambition to study mathematics.

He then entered Pembroke College Cambridge in 1956 from where he was awarded an M.A. It was during his second year of study that Michael Atiyah joined Pembroke College and had an instant beneficial effect on Kingman's studies [1]:-

[

Atiyah]supervised my pure mathematics: an education in itself. When I told him of my interest in probability and statistics, he arranged for me to be taught by Dennis Lindley, perhaps the best teacher of the subject for a generation. There were also quite a number of dreadful lecturers in Cambridge at that time, who had better remain nameless.

Kingman intended to undertake research with Lindley but he left Cambridge for Manchester. Kingman debated whether to follow Lindley to Manchester but decided to follow Lindley's advice and go to Oxford to undertake research under David Kendall. This he did but after Kingman had spent one year at Oxford, David Kendall moved to Cambridge where he was appointed Professor of Mathematical Statistics. At this stage Kingman returned to Cambridge but never finished his Ph.D. He did, however, continue collaborating with David Kendall during the next years that he spent in Cambridge.

In 1961 Kingman became a Fellow of Pembroke College, being a Smith's Prizeman in 1962. Also in 1962 Kingman was appointed as an assistant lecturer in mathematics. He was a visiting professor at the University of Western Australia in 1963, then he returned to Cambridge where he was promoted to lecturer in mathematics in 1964. It was in this year that he married Valerie Cromwell who was a historian.

In 1965 Kingman moved from Cambridge when he was appointed as reader in mathematics and statistics at the University of Sussex. Sussex was a very new university when Kingman was appointed there since it had only been founded a few years earlier. He said [1]:-

Sussex in the1960s was a very exciting place, alive with ideas and opportunities. My wife was teaching history there, and we made many friends across the whole range of subjects.

After only a year at Sussex, Kingman was promoted to a chair of mathematics and statistics in 1966. Two important books by Kingman were published in that year, namely *Introduction to Measure and Probability* (written jointly with S J Taylor) and *The Algebra of Queues.* His work on queues, particularly involving applications to traffic flows, have become an established area of operational research. The application to traffic queues had been amongst his earliest work with *The single server queue in heavy traffic* being published in 1961 and *On queues in heavy traffic* in the following year. Newell, reviewing this last mentioned paper, realised its significance at the time:-

Queueing theory, which has endured a long period in which people treated one example after another, is finally breaking out of its confinement to independent arrivals, service times, etc. The present paper represents a significant step in that direction.

In 1967 Kingman became a member of the International Statistical Institute. It was around this time that he was studying subadditive ergodic theory which in many ways turned out to be the most influential of all his mathematical achievements. The beginnings of the subject was a 1965 paper by J M Hammersley and D J A Welsh in which they made conjectured the subadditive ergodic theorem. Kingman succeeded in proving the theorem: he published *The ergodic theory of subadditive stochastic processes* in the *Journal of the Royal Statistical Society* in 1968 and *An ergodic theorem* in the *Bulletin of the London Mathematical Society* in the following year. Kingman gave a beautiful description of the development of the subject in his 1973 paper *Subadditive ergodic theory* published in the *Annals of Probability.*

Kingman was appointed as professor of mathematics at the University of Oxford in 1969 and he held this post until 1985. He explained why he was appointed to mathematics rather than statistics [1]:-

Statistics in Oxford in1969was frankly a mess. There was no professor of statistics, the only chair having been abolished some years before. ... I was appointed Professor of Mathematics to raise the profile of probability theory(but not statistics)in the Faculty of Mathematics.

Two years after his appointment, in 1971, his tremendous contributions to mathematical statistics were recognised when he was elected a Fellow of the Royal Society. This was the year in which he published another very important paper *Markov transition probabilities V* which gave complete characterization for the diagonal transition probabilities of a standard continuous-time Markov chain on a

countable state space. This paper gave, in some sense, a complete solution to a problem which Kingman had been studying since his paper *Markov transition probabilities. I* published in 1967. Between this first paper and the complete solution in 1971, Kingman had published several further contributions building up to the final elegant result: *Markov transition probabilities. II. Completely monotonic functions* (1967), *Markov transition probabilities. III. General state spaces* (1968), and *Markov transition probabilities. IV. Recurrence time distributions* (1968). This work was part of the theory of regenerative phenomena and, in addition to a number of other articles, Kingman published his classic text *Regenerative phenomena* in 1972.

While at Oxford he became a fellow of St Anne's College, holding this position from 1978 until 1985. During his time at Oxford, Kingman held several visiting appointments, in particular at the University of Western Australia in 1974 and the Australian National University in 1978. He held a number of important national positions during this time, such as chairman of the Science Board of the Science Research Council from 1979 to 1981 and then chairman of the Science and Engineering Research Council from 1981 until 1985.

In 1985 Kingman was appointed as Vice-Chancellor of the University of Bristol. Although he was at this stage in a position which no longer required him to retain his interests in mathematical statistics, Kingman certainly did not give up his mathematical interests. He was president of the Royal Statistical Society from 1987 to 1989 and president of the London Mathematical Society from 1990 to 1992. He continued his position as vice-president of the Institute of Statisticians which he had held from 1976 and he continued in this role until 1992. In 2001 Kingman left Bristol to become Director of the Isaac Newton Institute for Mathematics Sciences.

In addition to these mathematical activities, he undertook some less mathematical ones. He was a member of the board of the British Council between 1986 and 1991 and chairman of the Committee of Inquiry into the Teaching of the English Language in 1987-88. He also held a number of directorships such as IBM UK (1985-95) and Beecham Group (1986-89). He was associated with the British Technology Group, serving on the Council from 1984 until 1992 when he became a director.

We have commented above on several areas of Kingman's research as we were describing events in his life. Let us now sum up his mathematical contributions which were almost all in the area of mathematical statistics, more precisely stochastic analysis, random processes, regenerative phenomena and mathematical genetics. He wrote on the theory of Markov processes and published an important series of articles on Markov transition probabilities which we gave details of above. Among a large number of other topics to which he made major contributions were ergodic theorems, random walks and the theory of queues, in particular applying queuing theory to problems of traffic flow. He also wrote on a number of fascinating topics such as what happens to a piece of string when it is thrown.

In 1979 Kingman gave a series of lectures at Iowa State University on the contributions of mathematics to the study of genetic evolution. This was an area in which Kingman had made many important contributions himself and these are detailed in the lecture notes published as *Mathematics of Genetic Diversity* in the following year. Kingman discussed deterministic models and stochastic models relating to genetic evolution.

In 1993 Kingman published *Poisson processes* which provides a systematic treatment of the subject. A reviewer writes that the book:-

... fulfils the expectations one might have when a famous elder author writes a book on a classic topic. It gives the basic facts in a clear and lucid way. It is shown how the theory can be applied to interesting problems of astronomy, queuing and traffic etc., and these examples are studied very thoroughly and deeply, giving even the specialist new insights.

Kingman served as Director of the Isaac Newton Institute for Mathematical Sciences from October 2001 to September 2006. During that period he was the first N M Rothschild & Sons Professor of Mathematical Sciences. He was also the founding Chairman of the Statistics Commission, set up in 2000 to increase public confidence in UK National Statistics. He held this role until 2003.

Kingman has received many honours for his work in mathematics and statistics in addition to those, such as election as a fellow of the Royal Society, mentioned above. He was awarded the Junior Berwick Prize of the London Mathematical Society in 1967, the Guy medal in Silver from the Royal Statistical Society in 1981 and the Royal Medal of the Royal Society in 1983. He has been awarded honours by the universities of Sussex (1983), Southampton (1985), Bristol (1989), West of England and Hannover (1991), Queen's University, Ontario (1999). Kingman was knighted in 1985.

In July 2004, Kingman was awarded a D.Sc. by Brunel University.

**Article by:** *J J O'Connor* and *E F Robertson*

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