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One-sided local large deviation and renewal theorems in the case of infinite mean
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  • Published: April 1997

One-sided local large deviation and renewal theorems in the case of infinite mean

  • R. A. Doney1 

Probability Theory and Related Fields volume 107, pages 451–465 (1997)Cite this article

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Summary.

If {S n ,n≧0} is an integer-valued random walk such that S n /a n converges in distribution to a stable law of index α∈ (0,1) as n→∞, then Gnedenko’s local limit theorem provides a useful estimate for P{S n =r} for values of r such that r/a n is bounded. The main point of this paper is to show that, under certain circumstances, there is another estimate which is valid when r/a n → +∞, in other words to establish a large deviation local limit theorem. We also give an asymptotic bound for P{S n =r} which is valid under weaker assumptions. This last result is then used in establishing some local versions of generalized renewal theorems.

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  1. Statistical Laboratory, Department of Mathematics, University of Manchester, Manchester M13 9PL, UK, , , , , , GB

    R. A. Doney

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  1. R. A. Doney
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Received: 9 August 1995 / In revised form: 29 September 1996

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Doney, R. One-sided local large deviation and renewal theorems in the case of infinite mean. Probab Theory Relat Fields 107, 451–465 (1997). https://doi.org/10.1007/s004400050093

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  • Issue Date: April 1997

  • DOI: https://doi.org/10.1007/s004400050093

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  • Mathematics Subject Classification (1991): 60F10
  • 60J15
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