Double-transform Tauberian method for precise large deviations

TL;DR

This paper extends the Tauberian method to bivariate transforms, enabling precise large deviation analysis of complex stochastic processes with correlated increments and path constraints.

Contribution

It introduces a double-transform Tauberian approach for analyzing large deviations in processes within the domain of attraction of stable laws.

Findings

Derived precise large deviations for correlated random sums.

Analyzed path-dependent observables of constrained random walks.

Extended single-transform techniques to bivariate settings.

Abstract

In many stochastic models, the observables of interest are naturally encoded in double transforms (e.g., Laplace transforms) that couple spatial and temporal variables. Notably, the double transform often provides the only analytically tractable starting point for the study of processes with correlated increments or path constraints. We extend the Tauberian approach for precise large deviations of stochastic processes belonging to the domain of attraction of spectrally positive stable laws, previously developed for single-variable Laplace--Stieltjes transforms [9], to the bivariate setting. This methodology provides a direct route to asymptotic behaviour that is otherwise difficult to characterize using single-transform techniques. As illustrative examples, we derive precise large deviations for random sums with increments correlated to the stopping time and for path-dependent observables of random walks constrained to remain positive.