Recurrence and transience of operator semi-stable processes
Operator semi-stable distributions on the Euclidean space \(\mathbb{R}^d\) are studied by R. Jajte, W. Krakowiak, A. Łuczk, V. Chorny and others. A distribution \(\mu\) on \(\mathbb{R}^d\) is called operator semi-stable if there exist a sequence \(\{{\mathbf Y}_n: n=1, 2, \dots\}\) of i.i.d. (=independent identically distributed) random variables on \(\mathbb{R}^d\), a sequence of positive integers \(k_1< k_2< \dots\) satisfying \(k_{n+1}/ k_n\to r\) with some \(r\in [1,\infty)\), and a sequence of invertible linear operators \({\mathbf A}_n\) acting in \(\mathbb{R}^d\) and vectors \(b_n\) in \(\mathbb{R}^d\) for \(n= 1, 2,\dots\) such that the distribution of \({\mathbf A}_n ({\mathbf Y}_1+ {\mathbf Y}_2+ \cdots+ {\mathbf Y}_{k_n} )+b_n\) weakly converges to \(\mu\) as \(n\to \infty\). Since an operator semi-stable distribution on the Euclidean space \(\mathbb{R}^d\) is infinitely divisible, it induces a Lévy process \(\{{\mathbf X} (t)\}\) on \(\mathbb{R}^d\). We call Lévy processes on \(\mathbb{R}^d\) having operator semi-stable distributions at each time operator semi-stable processes. Here we mean by Lévy processes stochastically continuous processes with stationary independent increments starting at the origin. Our goal is to determine a non-degenerate operator semi-stable process whether recurrent or transient. Here we say that a Lévy process is non-degenerate if its distribution at each \(t>0\) is full. Our result is as follows: Let \(\{{\mathbf X}_t\); \(t\geq 0\}\) be a non-degenerate operator semi-stable process on the plane \(\mathbb{R}^2\). If \(\{{\mathbf X}_t\}\) is not Gaussian, then it is transient. Operator semi-stable processes on the line \(\mathbb{R}^1\) are semi-stable processes, and their classification is obtained. Also we know that for \(d>3\) all non-degenerate Lévy processes on \(\mathbb{R}^d\) are transient. Therefore our result completes classification of operator semi-stable processes into recurrence and transience.
scientific article; zbMATH DE number 851714
scientific article