The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks
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.
scientific article; zbMATH DE number 1023921
scientific article