an:01738012
Zbl 0996.60096
Sgibnev, M. S.
Stone's decomposition of the renewal measure via Banach-algebraic techniques
EN
Proc. Am. Math. Soc. 130, No. 8, 2425-2430 (2002).
00084751
2002
j
60K05
Stone's decomposition; renewal measure; asymptotic behavior; submultiplicative function; spread-out distribution; Banach algebra
Let \(F\) be a probability distribution on \(\mathbb R\) with positive mean \(\mu\) and let \(H\) be the corresponding renewal measure. \textit{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.
Valentin Topchii (Omsk)
Zbl 0147.16205