Second-order subexponential behavior of subordinated sequences (Q1881763)

Let \(X_1,\ldots ,X_n\) be independent identically distributed nonnegative integer-valued random variables with a distribution \(a(n)= P\{ X_j=n\}\), \(n\geq 0\). Denote \(S(n)=X_1+\cdots +X_n\). Suppose that \(N\) is another nonnegative integer-valued random variable with a distribution \(p(n)\). Then the random variable \(S(N)\) has the distribution \(b(n)=\sum_{k=0}^\infty p(k)a^{*k}(n)\) where \(a^{*k}\) denotes the \(k\)-fold convolution of \(a\). The sequence \(b\) is called subordinated with respect to \(a\). The authors study relations between the asymptotic behavior of \(a\) and \(b\). Typically, \(b(n)/a(n)\to EN\). Asymptotic estimates for \(b(n)-a(n)\to EN\) are given. Various possible applications are discussed.