Semi-stable stochastic processes

LAMPERTI 0. Introduction. An interesting chapter in modern probability theory began with the search for all the possible limit distributions for sums of independent, identically-distributed random variables. The result-the theory of the stable laws (see, for instance, l or 6) -generalizes and illuminates the original examples of normal convergence with which the problem originated. The purpose of this paper is to formalize and study an analogous situation in the theory of stochastic processes. To introduce the problem we use an example which has been very well known for a long time. Let Xt be a simple random walk on the line in which a moving particle starts from 0 at / = 0 and makes transitions at times t = nr, n an integer. The transitions are moves through a distance 8 to the right or left, each with probability 1/2. Then if t-»0 and ô->0, but ô2/t-»1, the process Xt converges to the Brownian motion (Wiener) process in the sense that the joint distribution of (Xtl, • ■ ■, Xtk) for the random walks converges to that for Brownian motion for all finite i-sets (2). It is quite natural to raise the following question: Which processes can similarly occur as limits upon subjecting a fixed stochastic process to infinite contractions of its time and space scales? It is essentially this class which we call "semi-stable. "The name is intended to suggest the analogy with the theory of stable laws, and is rendered more appropriate by the fact (see §2, Example 1 below) that a semi-stable process, if it is assumed to have independent increments, must actually be a stable one (3). 1. Foundations. As suggested by footnote 2, we shall consider two stochastic processes x¡ and y¡ to be the same if they have the same state space and finite-dimensional distribution functions; we indicate this by writing xt « yt. All the processes considered have states in Euclidean space of s dimensions, non-negative time parameter, and we assume throughout the continuity condition