Symmetric Random Walk?)

Let Xk, k= 1, 2, 3, • • -, be a sequence of mutually independent random variables on an appropriate probability space which have a given common distribution function F. Let Sn = Xi+ • • • +Xn, then the event lim inf | S„\ = 0 has probability either zero or one. If this event has zero chance, we say F is transient; in the other case, | 5„| tends to infinity almost surely, and F is called recurrent. The proofs of these assertions are in [l]. If F is symmetric, then transiency depends only on the tail of F. Theorem 1 gives a condition on the tail of the d.f. F which is necessary and sufficient for transiency. Let F and G be symmetric and F be less peaked than G, in the terminology introduced by Birnbaum. Theorem 4 shows that, if F is unimodal, then the recurrence of F implies the recurrence of G. The unimodality condition cannot be entirely removed as an illuminating example shows. Unimodal distribution functions play a central role and Theorem 4 shows the connection between this class and the uniform distributions via the representation of Khinchin. The condition in Theorem 1 can be very much simplified in case F is unimodal, this is done in §5. Finally the results are shown to extend to the higher dimensional case. 1. Preliminaries. It is well known that a distribution function (d.f.) F is transient if and only if (1.1) is finite.