Rates of Convergence in Laplace’s Integrals and Sums and Conditional Central Limit Theorems

Vassili N Kolokoltsov

doi:10.3390/math8040479

Abstract

We obtained the exact estimates for the error terms in Laplace’s integrals and sums implying the corresponding estimates for the related laws of large number and central limit theorems including the large deviations approximation.

Highlights

The Laplace integrals find applications in numerous problems of mathematics and applied science, and the literature on these integrals is abundant
Let us mention the applications in statistical physics, see e.g., [1] or Lecture 5 in [2], in the pattern analysis [3], in the large deviation theory [4,5,6], where it is sometimes referred to as the Laplace–Varadhan method, in the analysis of Weibullian chaos [7], in the asymptotic methods for large excursion probabilities [8], in the asymptotic analysis of stochastic processes [9], and in the calculation of the tunneling effects in quantum mechanics and quantum fields, see [10,11]
The majority of research on this topic is devoted to the asymptotic expansions, or even, following the general approach to large deviation of Varadhan, just to the logarithmic asymptotics, see [15]

Summary

Introduction

The Laplace integrals find applications in numerous problems of mathematics and applied science, and the literature on these integrals is abundant. In the present paper, following the recent trend for the searching of the best constants for the error term in the central-limit-type results, see [16] and references therein, we are interested in exact estimates for the main error term of the Laplace approximation. This approach to Laplace integrals was initiated by the author in book [9] (Appendix B), where the stress was on the integrals with complex phase.

Phase Minimum Inside the Domain of Integration

LLN and CLT for Internal Minima of the Phase

Phase Minimum on the Border of the Domain of Integration

LLN and CLT for Minima on the Boundary

Laplace Sums with Error Estimates

Application to LLN and CLT of Large Deviations