Applied Probability in Great Britain

T he subject of probability was, of course, motivated initially by its applications. However, there have been earnest attempts to convert it into a branch of pure mathematics, a development suggested by the axiomatic basis of the theory and reinforced by a preoccupation with possible measure-theoretic pathologies. Excesses in this direction led to a reaction, marked by the appearance of the journals Journal of Applied Probability (1963) and Advances in Applied Probability (1968) in Britain and Annals of Applied Probability (1991) in the United States, plus, of course, explicit mention of applications in the titles of the key Russian and German probability journals. Natural pragmatism had in fact kept the development of the subject fairly continuous in Britain-see the opening quotation, whose reference to England should, as often, be taken as meaning Britain (or even the Anglo-Celtic population of the entire British Isles). The journals JAP and AAP thus found a natural home there, although the foundation and operation of the Applied Probability Trust with which they are associated owes everything to Joe Gani's untiring energy. (See Gani 1988 for his own account.) This survey is largely restricted to the post-war (World War II) surge of development, which began with the activity of people such as Bartlett, Daniels, D. G. Kendall, and Moran. However, there is, of course, a considerable body of classic earlier work. Much of this has a clear biological slant, such as the studies by Galton and Watson of surname extinction and branching processes, Fisher's studies of genetic models (notably Fisher 1937, which still reverberates in the literature), and the treatments by Kermack and McKendrick (1927) of epidemic models. The war itself gave impetus to stochastic modelling of an operational research character, such as E. A. Milne's study of the optimal distribution of fragmentation of anti-aircraft shells. The view taken in this article is a personal one rather than an attempt at an exhaustive enumeration, although fairness has been a goal throughout. Coverage of some more recent applications has not been attempted, despite their interest and substance, e.g., finance and artificial neural nets. By its nature, the survey is concerned with models and not with inference. So, for instance, the statistical methods of time series analysis (to which British authors have contributed prolifically) are excluded. The stochastics of time series models are indeed interesting but have long been clarified in the familiar linear stationary case (the only distinctive British contribution being perhaps that of Yule 1927, with his suggestion of the linear autoregression). The author is aware that he is over-represented in the reference list. This is a consequence purely of a spread of interests-one of the few favourable to him, some would say.