On the asymptotic behavior of the harmonic renewal measure (Q1266783)

Let \(X_1,X_2,\dots\) be independent random variables with distribution function \(F\) and negative mean, let \(S_0=0\), \(S_n=\sum^n_{k=1}X_k\) be the associated partial sums. Then the harmonic renewal measure \(U\) is given by \(U(A)= \sum^\infty_{n=1} n^{-1}P (S_n\in A)\). The main result of the paper deals with the connection between the behaviour of \(x\to 1-F(x)\) and \(x\to U([x,\infty))\) as \(x\to\infty\).