It was fortunate that a mathematician of such high calibre and with such an excellent background in relevant areas of pure mathematics was entering the field of probability and statistics at this time. Following the path-breaking work of A. N. Kolmogorov and the Russian school, the subject was entering its most exciting phase -the transition from a ragbag of special cases and non-rigorous arguments to the mature mathematical field of today. Doob was one of the prime movers in this transformation.

Joseph Leo Doob was born in Cincinnati in 1910. He studied at Harvard University, taking his PhD there in 1932. In 1935 he joined the mathematics staff at the University of Illinois, Urbana-Champaign, where he remained for the rest of his career.

Doob's greatest single achievement is his classic book, *Stochastic Processes* (1953). A stochastic- or random-process is a mathematical model of a situation in which randomness evolves with time. Since chance, or uncertainty, dominates many phenomena, and the real world evolves with time, stochastic processes are both very general and ubiquitous.

Doob's book was the first systematic study of them to harness the full power of modern mathematics. It had an immediate and decisive influence.

Doob's distinctive contribution was his development of a particular kind of stochastic process -- martingales. These capture the idea of a complete absence of predictability or trend, and so are useful for modelling the unpredictable disturbances, or "noise", in nature.

Doob laid bare their fundamental properties -- broadly speaking, he showed that they share many of the cancellation or averaging properties of that most basic example, the repeated tossing of a fair coin. He could not have foreseen how martingales would dominate the mathematics of financial derivatives -- the options and other products behind the explosive recent growth of the financial services industry.

Doob's second book, *Classical Potential Theory and its Probabilistic Counterpart* (1984) -- written in his retirement -- unified his life's work on probability with the potential theory he had studied in mathematics as a young man.

His last book, *Measure Theory* (1993), he wrote partly to give his own approach to the measure theory that underpins probability theory, partly because he had discovered the joys of wordprocessing. He had typed all seven drafts of his first book, some 650 pages, and was delighted that life had now become so much easier in this respect.

Doob was widely honoured. He was elected a member of the US National Academy of Sciences in 1957 and of the American Academy of the Arts and Sciences in 1965. He received the National Medal of Science in 1979 from President Carter. He was President of the Institute of Mathematical Statistics (1950) and the American Mathematical Society (1963-64). He was awarded the Career Prize by the American Mathematical Society in 1984.

Doob's father, Leo Dub, was a Czech Jew who emigrated to the US, where he changed his name (it means "oak" in Czech) to Doob when he tired of being called "dub".

Doob and his wife Elsie Field married when they were very young -- "We were kinda immatoor," he would say. It was a very happy marriage, which lasted more than 50 years till Elsie, a doctor specialising in birth control, died in 1991; they had three children.

When he became famous, Doob received numerous offers of jobs at the most prestigious universities. He would always decline, saying that he could not leave Urbana-Champaign because it had an excellent hardware store; Doob's impressive mathematical library was housed on an array of shelves that he put up himself.

As he put it, he was charmed by the small-town atmosphere of Urbana as soon as he arrived, and never wanted to leave. But also, Doob was always grateful to the University of Illinois for giving him his first job, at a time when Jews were not always treated fairly by universities when hiring.

Doob was sensitive to injustice, and he was proud of having persuaded a leading US university to allow one of his pupils (David Blackwell, perhaps the most distinguished black American mathematician) access to its previously segregated library.

Doob was outspoken, completely unpretentious, had broad interests and was very good company. He loved the outdoors -- hiking, canoeing and fishing. For more than 60 years, he was an enthusiastic member of the famous Saturday Hike, where in all weathers the group would drive out to the country, hike for hours, and then settle down to the main business, talking for hours over a camp fire and a barbecue.

Joe Doob will be remembered for his life's work, which included scientific books and papers. He will also be remembered for his personal qualities: bluff kindness, good humour and wit, and a complete lack of personal ambition, combined with gritty dedication to ahighly technical field of study over a working lifetime of 70-odd years.

His life alone disproves the popular myth that mathematics is a young man's game.

Joseph Doob is survived by two sons and a daughter.

Joseph Doob, mathematician, was born on February 27, 1910. He died on June 7, 2004, aged 94.

© The Times, 2004