Closure properties of O-exponential distributions

Sufficient conditions are found under which randomly stopped sums, randomly stopped maximums, and randomly stopped maximums of sums are distributed according to O-exponential laws. It is supposed that the basic random variables {ξ1,ξ2,…} are real valued and not necessarily identically distributed, whereas the counting random variable η is integer valued, nonnegative, and not degenerate at zero.