Biographical Memoirs Volume 90 (2009) / Chapter Skim
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WILLIAM FELLER
WILLIAM FELLER
Pages 70-85
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Photograph by Orren Jack Turner, Courtesy Princeton University Library.
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and received the equivalent of a master of science degree there. Feller then entered the University of Göttingen in 1925 and completed his doctorate with a thesis titled "Uber algebraisch rektifizierbare transzendente Kurven." His thesis advisor was Richard Courant.
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obtained necessary and sufficient conditions for the central limit theorem of probability theory. This was a culmination of earlier work of DeMoivre, Laplace, and Liapounov.
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In 1931 a paper of Kolmogorov on analytic methods in probability theory discussed the differential equations satisfied by the transition probabilities of continuous time parameter Markov processes (random processes with the property that past and future of the process are conditionally independent given precise knowledge of the present) .2 The conditional probability that a system at time t at location x will at time τ>t be less than or equal to y is given by a function F(t,x,τ,y)
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f] =0 ∂y ∂y The case of purely discontinuous or jump Markov processes in continuous time was also examined and under appropriate conditions an integrodifferential equation is shown to be satisfied by the constructed transition probability function of the Markov process.
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An elegant paper (1949,1) written jointly with Paul Erdös and Harry Pollard used elementary methods to establish limit behavior of transition probabilities for countable state discrete time Markov chains under appropriate conditions (a result obtained earlier by Kolmogorov using arguments that were more elaborate)
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An algebraic treatment is given for finite state Markov chains. Finally birth and death processes are introduced and considered as examples of countable state continuous time parameter processes.
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I recall some impressions from my own days as a graduate student in the late 1940s at Cornell, where I took most of my courses in stochastics with Feller as a lecturer. Though I completed my doctorate with Mark Kac as adviser, I had an overwhelming impression of Will Feller as a man of supreme enthusiasm and occasional exaggeration that at times required some modification.
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A corresponding Markov diffusion process was determined by boundary conditions that might differ from those conventionally dealt with in the standard theory of differential equations. The boundary conditions could be regarded as a restraint on the class of functions on which L operated that in turn made the restrained L the infinitesimal generator of a properly
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Feller also carried out analyses of countable state continuous time Markov chains with stationary transition probabilities. Here the transition probabilities are given by a matrix-valued function P(t)
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Infinitely divisible distributions follow in the next chapter. A host of additional topics follow in the remaining chapters of the book: Markov processes and semigroups, renewal theory, random walks on the real line, characteristic functions, expansions related to the central limit theory, the Berry-Esséen theorem on the error term in the central limit theorem, large deviations, and aspects of harmonic analysis.
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A colleague of many years at Cornell University and Rockefeller University, also a remarkable researcher in probability theory, Mark Kac, said of Feller: Feller was a man of enormous vitality.
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Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Springer, 1933.
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42:301-312. 1939 Die Grundlagen der Volterraschen Theorie des Kampfes ums Dasein in wahrscheinlichkeitstheoretischer Behandlung.
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47:631-638. 1948 On the Kolmogorov-Smirnov limit theorems for empirical distribu tions.
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18:343-356. 1970 On the oscillations of the sums of independent random variables.
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