On the Kolmogorov-Prokhorov theorem on the existence of expectations of random sums
Let \(\xi_1,\xi_2,\dots\) be independent, not necessarily i.i.d., random variables (r.v.) and \(v\) an integer valued r.v. independent of the future, i.e. \(\{v\leq n\}\) is independent of \(\xi_i\), \(I> n\). Let \(S_v=\xi_1+\cdots+ \xi_v\). The paper discusses generalizations of the Kolmogorov-Prokhorov theorem on \(ES_v\) and derives bounds on \(Eg(|S_v|)\) for suitable \(g\) satisfying \[ g(x)= \int^x_0 p(u)\,du,\tag{3} \] \(x> 0\), with \(p\) nonnegative, right continuous and nondecreasing and \[ g(x, y)\leq Cg(x)g(y),\;x,\;y> 0.\tag{4} \] It is shown that then \[ E\{g(|S_v|)/p(v)\}\leq C \sum^\infty_{k=1} P(v\geq k) Eg(|\xi_k|). \] When the \(\xi_i\) are i.i.d. the right-hand side becomes \(CEg(|\xi_1|)Ev\). When \(v,\xi_1,\xi_2,\dots\) are independent a sharper bound is derived giving corollaries for i.i.d. \(\xi_i\). An example is given showing that (4) cannot always be missed.
scientific article; zbMATH DE number 5621359