Random integral representations for classes of limit distributions similar to Lévy class L0, II

Abstract

Let ξ(t) and η(t) be two stochastic processes such that ξ has stationary independent increments and ξ(0) = 0 a.s. Suppose that for each 0 < t ≤ 1, with ξ(tβ) independent of η(t) and a fixed parameter β ∈ (−2, 0). It is shown that ξ(1) satisfies the above equation if and only if ξ(1) is a sum of two independent r.v.’s: strictly stable one with the exponent – β and the one given by a random integral where Y has stationary independent increments and E [|| Y(1)||-β] < ∞.