The maximum of sums of stable random variables

for random variables Xi which have a stable distribution. In case the X{ are normally distributed the limiting distribution of max Sk has been known for some time and turns out to be the truncated normal distribution; the limiting distribution of max \Sk\ is also known and is a theta function (cf. [l]). The same results obtain when the Xi are not necessarily identically distributed but merely such that the central limit theorem applies to them (cf. [2]). In these cases the limiting distributions are the same as the distributions of the corresponding functionals for the Wiener stochastic process which can be, in turn, formulated as boundary value problems for the simple diffusion equation, whose solutions are classical. But in the case the X, belong to the domain of attraction of a stable law other than the normal, the problem cannot be reduced to a diffusion equation, and none of the standard methods seems to work. In one particular case a solution has been given by Kac and Pollard [3]. They found the limiting distribution of max |5*| when the Xi are identically distributed Cauchy variables. However their method failed to yield anything for the one-sided maximum, max Sk. In this paper we find the limiting distribution of max Sk when the variables Xi have a symmetric stable distribution of index y, 0<7^2. Corresponding results could no doubt be obtained under the condition the Xi merely belong to the domain of attraction of a stable law, using the present method, but we restrict our treatment to the present case for simplicity. The problem of determining the limiting distribution of max \Sk\ ior general stable variables is still apparently open. 2. An initial reduction. The results of this paper are based upon, and made possible by, the following basic result of Spitzer [4]. Let Xi, X2, • • • be