Asymptotic and Non-asymptotic Estimates for Multivariate Laplace

Maria Rosaria Formica,Leonid Sirota,Eugeny Ostrovsky

doi:10.19139/soic-2310-5070-806

Abstract

We derive bilateral asymptotic as well as non-asymptotic estimates for the multivariate Laplace integrals. Furthermore, we provide multidimensional Tauberian theorems for exponential integrals.

Highlights

The one-dimensional case d = 1 was considered in [6, 22, 23]; a preliminary result may be found in [25]
In this paper we provide asymptotical as well as non-asymptotical upper and lower estimates of the Laplace integral I[ζ](λ) = I(λ), for all sufficiently large values of the real vector parameter λ = ⃗λ ∈ Rd+, d = 1, 2, 3, . . ., for Λ(λ) ≥ 1 and when Λ → ∞; we obtain direct es√timations of I(λ) assuming, its convergence for all the sufficiently large values of the parameter |λ| := (λ, λ)
We will generalize the main results obtained in the articles [22, 23, 25], where are described some applications of these estimates, in particular, in the probability theory

Summary

Introduction

The function K(ε), ε > 0, defined by (9), satisfies the following estimate Let us impose the following condition on the function φ(·) : Where m = const > 0, | · | is the ordinary Euclidean norm (or an other arbitrary non-degenerate vector one) and L = L(r), r ≥ 1, is some positive continuous slowly varying function as r → ∞, and we suppose

Results

Conclusion