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Increments of Erdős-Rényi type of cumulative renewal processes. (Accroissements de type Erdős-Rényi du processus de renouvellement cumulé.) (French) Zbl 0780.60029

Summary: Let \(\bigl\{Z(t)\bigr\}_{t\geq 0}\) be the cumulative process associated with the sequence of the positive, independent random variables \(\{Y_ i\}_{i\geq 1}\) and to \(\bigl\{N(t)\bigr\}_{t\geq 0}\), where \(N(t)\) is a renewal process associated with the sequence \(\{Y_ i\}_{i\geq 1}\) of interoccurrence-times which are positive i.i.d. random variables. We study the asymptotic behaviour of the increments of \(Z(t)\) defined by \(Z(t)=\sum^{N(t)}_{j=1}Y_ j\) if \(N(t)\geq 1\), \(Z(t)=0\) if \(N(t)=0\), of the form \(D(T,K)=\sup_{0\leq t\leq T-K}\bigl\{(Z(t+K)-Z(t))/K\bigr\}\) when \(K=K_ T\) satisfies suitable conditions of growth and regularity [see the full form of the Erdős-Rényi theorem in P. Deheuvels, L. Devroye and J. Lynch, Ann. Probab. 14, 209-223 (1986; Zbl 0595.60033)].

MSC:

60F15 Strong limit theorems

Citations:

Zbl 0595.60033
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