A renewal theorem of Blackwell type
Let \(S_ n\) be the sum of the first n terms of a sequence of i.i.d. random variables with distribution F having positive finite mean \(\mu\), and let \(G(x)=\sum^{\infty}_{n=1}a(n)P(S_ n\leq x)\), where a(x) is a positive function which varies regularly at infinity with exponent \(\alpha\). It is shown that if \(\alpha>-1\) and F is non-lattice then \(G(x+h)-G(x)\sim h\mu^{-\alpha -1}a(x)\), for \(h>0\), as \(x\to \infty\). This asymptotic relation holds also for \(\alpha\leq -1\) under some additional assumptions.
scientific article; zbMATH DE number 3854162