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A renewal theorem of Blackwell type. (English) Zbl 0537.60087
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.
Reviewer: T.Mori

##### MSC:
 60K05 Renewal theory
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