SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS Y. K. Chan has done much to constructivize the theory of stochastic processes. His paper [lo] unifies the two classically distinct cases of the renewal theorem into one constructive result. His paper [12] contains the following theor em: THEOREM. Let p and p be probability measures on R, and fl and f2 their characteristic functions (Fourier transforms). Let g be a con- tinuous function on R, with lgl s 1. Then for every c 0 there exist 6 0 and 9 0, depending only on c and the moduli of continuity of fl, f2, and g, such that whenever If - f21 8 on [- 9. 81 . 1 A simple corollary is Levy's theorem. that if {pn3 is a sequence of probability measures on W, whose characteristic functions {f 3 converge n uniformly on compact sets to some function f , then p n converges weakly to a probability measure p whose characteristic function is f Levy's theorem is classically an important tool for proving converg- ence of measures. Chan shows that this is also true constructively, by using it to get constructive proofs of the central limit theorem and of the Levy- Khintchine formula for infinitely divisible distributions. Chants papers 191 and [ 111 are primarily concerned with the problem of constructing a stochastic process. In [9] he gives a constructive version of Kolmogorov's extension theorem. In [ 111 , he constructivizes the classical derivation of a time homogeneous Markov process from a strongly continuous semi-group of transition operators. In addition he proves Ionescu Tulcea's 1 theorem and a supermartingale convergence theorem. H. Cheng [13] has given a very pretty version of the Riemann mapping theorem and Caratheodory's results on the convergence of mapping functions. He defines a simply connected proper open subset U of the complex plane C to be mappable relative to some distinguished .point z of U if for each 0 @ 0 there exist finitely many points z l r . . . , ~ in the complement of U n such that any continuous path beginning at z and having distance 2 c from 0 each of the points z l,...,z n lies entirely in U . He shows that mppability is necessary and sufficient for the existence of a mapping function. He goes on to study the dependence of the mapping function on the domain, by defining
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