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Bulletin of the London Mathematical Society 2001 33(1):100-108; doi:10.1112/blms/33.1.100
© 2001 by London Mathematical Society
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© London Mathematical Society

A Local Limit Theorem for Moderate Deviations

R. A. Doney

Department of Mathematics, University of Manchester Oxford Road, Manchester M13 9PL; e-mail: rad{at}ma.man.ac.uk

Received 24 May 1999. Revision received 2 December 1999. Revision received 1 February 2000.

The purpose of this note is to establish a uniform estimate for the mass function P(Sm = y) of an integer-valued random walk when y -> {infty} and Formula where µ isthe mean of the step distribution. (The local central limit theorem provides such an estimate when (Formula is bounded.) The assumptions are that the mass function p of the step distribution is regularly varying at {infty} with index –{kappa}, where {kappa} > 3, and that Formula for some {kappa}' > 2. From this result, a ratio limit theorem is derived, and this in turn is applied to yield some new information about the space–time Martin boundary of certain random walks.


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