Asymptotic behavior of a generalized renewal measure under weak moment conditions (Q1597965)

Let \(X_i\), \(i=1,2,\dots\), be independent identically distributed random variables and let \(S_k\) be their partial sums. The author studies the asymptotic behavior of the generalized renewal measure \[ \Phi_n(\cdot)=\sum_{k=0}^{\infty}n\cdot(n+k-1)!{\mathbb P}(S_k\in \cdot)/k!, \] where \(n\geq 1\) is an integer, under the additional assumptions that \({\mathbb E}|X_1|^m<\infty\) and \({\mathbb E}|X_1|^{m+1}=\infty\) for some \(m\leq n\).