On the random walk and Brownian motion

Abstract

(Wiener process) X(t), 0 t < oo, X(0) = 0, and on the other a classical random walk S(n) = ,'J= Xi, 1 _ n < co, where X1, X2, * * * is a sequence of Bernoulli trials probability 1/2 for Xi= +1 and for Xi= -1. It is well known that for the sequence of processes Rk(t), 1< k <oo, defined by the formula Rk(t) = 2-kS([22kt]), 0 < g < 00, the joint distributions converge weakly in k to the corresponding joint distributions of X(t). It is also well known [7] that the difference equation satisfied by the transition function of Rk(t) goes over into the diffusion equation satisfied by the transition function of Brownian motion. Connections of the type illustrated here between the random walk and the Brownian motion involve the derivation of analytical properties of the latter as limits of analogous properties of the former, independently of the probability spaces on which the processes are defined. However, it is also possible to define, as in the present paper, approximating sequences of random walks together a limit Brownian motion process on a single probability space, in such a way that the customary limit theorems appear as consequences of probability 1 convergence on this common space, and this method can have certain advantages. For example, when the type of convergence on this space is uniform convergence of path functions in finite time intervals probability 1, as it is to be here, it follows that the random walks provide a constructive definition of the Brownian motion (that is, of its space of path functions, field of measurable sets, and probability measure). This paper is derived from part of the author's Ph.D. thesis, written the help and guidance of Professor William Feller. Sincere thanks are rendered to him and to Professor H. Trotter for their assistance. The mappings Mk, on which the paper is based, are due to them. The first stage in the construction is to define a sequence of random walks Rk(t) of the type mentioned in the introduction for each k, but on separate probability spaces, and to combine these spaces in a single over-all space. We shall use a.s. to abbreviate almost surely or with probability one. DEFINITION 1. Let ak= 2-2k, /k= 2-1, where k is a non-negative integer. The set Qk of Rk-paths is the set of all functions Wk from { mak: m a non-negative integer} onto {n0k: n an integer}, such that Wk(0) 0 and Wk((M+l)?ak)