A local limit theorem for random walk maxima with heavy tails

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Abstract

For a random walk with negative mean and heavy-tailed increment distribution F, it is well known that under suitable subexponential assumptions, the distribution π of the maximum has a tail π(x,∞) which is asymptotically proportional to xF(y,∞)dy. We supplement here this by a local result showing that π(x,x+z] is asymptotically proportional to zF(x,∞).

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Acknowledgements

After the submission of this work, Professor V.V. Kalashnikov sadly passed away. His devotion as well as his inspiration and enthusiasm live on in the memories of the coauthors of this paper.

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