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Stone’s decomposition of the renewal measure via Banach-algebraic techniques. (English) Zbl 0996.60096
Let $$F$$ be a probability distribution on $$\mathbb R$$ with positive mean $$\mu$$ and let $$H$$ be the corresponding renewal measure. C. Stone [Ann. Math. Stat. 37, 271-275 (1966; Zbl 0147.16205)] showed that, if for some $$m\geq 1$$ $$m$$-times convolution of $$F$$ has a nonzero absolutely continuous component, then there exists a decomposition $$H=H_1+H_2$$, where $$H_2$$ is a finite measure and $$H_1$$ is absolutely continuous with bounded continuous density $$h(x)$$ such that $$\lim_{x\to+\infty}h(x)=\mu^{-1}$$ and $$\lim_{x\to-\infty}h(x)=0$$. A lot of estimations are based on the representation of $$H$$ under some additional assumptions. Stone’s decomposition is proved by using Banach-algebraic techniques. The method allows to extract detailed information about the asymptotic properties of the terms $$H_1$$ and $$H_2$$. Under some additional restrictions of submultiplicative type, estimates of the rate of convergence in the key renewal theorem are obtained.

##### MSC:
 60K05 Renewal theory
Full Text:
##### References:
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