The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks
type: Item
instance of: scholarly article
title: The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks
zbMATH Open document ID: 0878.60023
DOI: 10.1007/BF02104660
published in: Siberian Mathematical Journal
publication date: 1997-11-21T00:00:00Z
review text: Let \(\xi\) be a nondegenerate random vector in the \(d\)-dimensional Euclidean space \(R^d\), \(d\geq 1\), with distribution \(F\). The Laplace transform of \(\xi\) is defined by \(\varphi(\lambda)=\int e^{\langle\lambda, v\rangle}F(dv)\), \(\lambda\in R^d\), where \(\langle\cdot,\cdot\rangle\) is the inner product in \(R^d\). The deviation function (in the context of large deviations) for \(\xi\), or its distribution \(F\), is defined to be the Legendre transform \(\Lambda(\alpha)\) of the function \(A(\lambda)=\ln\varphi(\lambda)\): \[ \Lambda(\alpha)=\sup_\lambda\{\langle\lambda,\alpha\rangle- \ln\varphi(\lambda)\},\quad \alpha\in R^d. \] The deviation function plays a crucial role in the investigation of large deviation probabilities for sums \(S_n=\xi_1+\xi_2+\cdots+\xi_n\), where \(\xi_i\) are independent and distributed as \(\xi\). The second deviation function (second rate function) is defined by \[ D(\alpha)=\inf_{s>0}(\Lambda(s\alpha)/s),\quad \alpha\in R^d. \] This function is important in the study of the renewal function \(H(V)=\sum_{n=1}^\infty {\mathbf P}(S_n\in V)\). The large deviation principle for \(H(V)\) is established and asymptotic expansions are found for \(H(tV)\) as \(t\to\infty\). Another goal of the article is to study the properties of the second deviation function \(D(\alpha)\) necessary for the analysis of \(H(V)\) and the related boundary-hitting functions.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 1023921
zbMATH Keywords: large deviation principle
zbMATH Keywords: deviation function
zbMATH Keywords: Legendre transform
zbMATH Keywords: Cramér-type condition
zbMATH Keywords: renewal function
zbMATH Keywords: second deviation function
author: A. A. Borovkov
author: Anatoli Mogulskii
reviewed by: Birutė Kryžienė
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02104660
OpenAlex ID: W2051871795
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description: scientific article; zbMATH DE number 1023921