On subordinated distributions and generalized renewal measures (Q1819455)

Let \(X_ 1,X_ 2,..\). be a sequence of independent, identically distributed random variables with partial sums \(S_ 0=0\), \(S_ n=X_ 1+...+X_ n\). We investigate the behaviour of \(\sum^{\infty}_{n=0}a_ nP(S_ n\in x+A)\) as \(x\to \pm \infty\), where \(a_ 1,a_ 2,..\). is a sequence of nonnegative numbers and \(A\subset {\mathbb{R}}\) is a fixed Borel set. The special case \(a_ n=1\), \(A=(-\infty,0]\), gives the renewal function where our method leads to an expansion down to terms of order \(P(X_ 1\in [x,x+1])\). Other applications are to harmonic renewal measures and infinitely divisible distributions.