Blackwell-type theorems for weighted renewal functions

A. A. Borovkov^ab, K. A. Borovkov<sup>c

a</sup> Sobolev Institute of Mathematics, Novosibirsk, Russia
^b Novosibirsk State University, Novosibirsk, Russia
^c The University of Melbourne, Parkville, Australia

Abstract: We establish Blackwell-type theorems for weighted renewal functions under much weaker conditions, as compared to the available, on the weight sequence and the distributions of the jumps in the renewal process. The proofs are based on using integro-local limit theorems and large deviation bounds. For the jump distribution, we consider conditions of the four types: (a) it has finite second moment, (b) it belongs to the domain of attraction of a stable law, (c) its tails are locally regularly varying, and (d) it satisfies the moment Cramér condition. In cases (a)–(c), the weights are assumed to satisfy a broad regularity condition on their moving averages, whereas in case (d) the weights can change exponentially fast.

Keywords: weighted renewal function, Blackwell theorem, integro-local limit theorems, Stone–Shepp theorems, large deviation probabilities, locally constant functions, regular variation.

UDC: 519.21

Received: 07.02.2014

 English version: Siberian Mathematical Journal, 2014, 55:4, 589–605

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