Premium
A Local Limit Theorem for Moderate Deviations
Author(s) -
Doney R. A.
Publication year - 2001
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/33.1.100
Subject(s) - mathematics , random walk , central limit theorem , limit (mathematics) , bounded function , uniform limit theorem , distribution (mathematics) , combinatorics , integer (computer science) , space (punctuation) , function (biology) , donsker's theorem , mean value theorem (divided differences) , mathematical analysis , discrete mathematics , picard–lindelöf theorem , statistics , danskin's theorem , fixed point theorem , linguistics , philosophy , evolutionary biology , computer science , biology , programming language
The purpose of this note is to establish a uniform estimate for the mass function P( S m = y ) of an integer‐valued random walk when y → ∞ and( y − m μ ) / m → ∞ where μ isthe mean of the step distribution. (The local central limit theorem provides such an estimate when ( ( y − m μ ) / mis bounded.) The assumptions are that the mass function p of the step distribution is regularly varying at ∞ with index −κ, where κ > 3, and that∑ n = 0 ∞n κ ′ p ( – n ) < ∞ for some κ′ > 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.