Random convolution of different o-exponential distributions

Random convolution of different O-exponential distributions. Assume that ξ1, ξ2, ... are independent non-negative random variables with distribution functions Fξ1, Fξ2, .... Suppose that η is a non-negative non-degenerate at zero integer-valued random variable independent of {ξ1, ξ2, ...}. Assume that S0 = 0, Sn = ξ1 + ξ2 +...+ ξn, n ∈ N, and Sη = ξ1 + ξ2 + ... + ξη. The main goal of this paper is to consider the conditions for the random variables ξ1, ξ2, ... and η under which the distribution function FSη remains in the class OL