Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Precise large deviations through a uniform Tauberian theorem

Giampaolo Cristadoro, Gaia Pozzoli

TL;DR

This work develops a uniform Tauberian framework for bilateral Laplace-Stieltjes transforms to obtain precise large deviation principles in the basin of attraction of spectrally positive stable laws. By coupling regular variation with a uniform control on transforms across a family indexed by $t$, the authors extend classical results beyond Cramér's condition and address random sums and randomly stopped sums with both finite and infinite mean stopping times. Central contributions include a uniform Tauberian theorem (Theorem tauberian) and a corresponding large-deviation principle (Theorem LDP), plus extensive applications to sums with independent increments and to sums stopped at random times, including realistic models with memory kernels and CTRW-type dynamics. The results provide a unified, adaptable approach that captures non-exponential tail decay and can incorporate non-analytic characteristic-function behavior when Cramér’s condition fails, with potential impact on renewal theory, risk, and anomalous diffusion modeling.

Abstract

We derive a large deviation principle for families of random variables in the basin of attraction of spectrally positive stable distributions by proving a uniform version of the Tauberian theorem for Laplace-Stieltjes transforms. The main advantage of this method is that it can be easily applied to cases that are beyond the reach of the techniques currently used in the literature. Notable examples include large deviations for random walks with long-ranged memory kernels, as well as for randomly stopped sums where the random time $N$ is either not concentrated around its expectation or has an infinite mean. The method reveals the role of the characteristic function when Cramér's condition is violated and provides a unified approach within regular variation.

Precise large deviations through a uniform Tauberian theorem

TL;DR

This work develops a uniform Tauberian framework for bilateral Laplace-Stieltjes transforms to obtain precise large deviation principles in the basin of attraction of spectrally positive stable laws. By coupling regular variation with a uniform control on transforms across a family indexed by , the authors extend classical results beyond Cramér's condition and address random sums and randomly stopped sums with both finite and infinite mean stopping times. Central contributions include a uniform Tauberian theorem (Theorem tauberian) and a corresponding large-deviation principle (Theorem LDP), plus extensive applications to sums with independent increments and to sums stopped at random times, including realistic models with memory kernels and CTRW-type dynamics. The results provide a unified, adaptable approach that captures non-exponential tail decay and can incorporate non-analytic characteristic-function behavior when Cramér’s condition fails, with potential impact on renewal theory, risk, and anomalous diffusion modeling.

Abstract

We derive a large deviation principle for families of random variables in the basin of attraction of spectrally positive stable distributions by proving a uniform version of the Tauberian theorem for Laplace-Stieltjes transforms. The main advantage of this method is that it can be easily applied to cases that are beyond the reach of the techniques currently used in the literature. Notable examples include large deviations for random walks with long-ranged memory kernels, as well as for randomly stopped sums where the random time is either not concentrated around its expectation or has an infinite mean. The method reveals the role of the characteristic function when Cramér's condition is violated and provides a unified approach within regular variation.
Paper Structure (19 sections, 10 theorems, 101 equations)

This paper contains 19 sections, 10 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Classical large deviations: Cramer's condition
  3. Subexponentiality and the big-jump domain
  4. Regular variation and Tauberian theorems
  5. Statement of the main results
  6. Applications
  7. Sums with independent increments
  8. Sums of a random number of independent increments
  9. A couple of explicit examples
  10. Processes $(N(t))_{t\geq 0}$ with finite mean
  11. Processes $(N(t))_{t\geq 0}$ with infinite mean
  12. Proofs
  13. Proof of Theorem \ref{['thm:tauberian']}
  14. Proof of Theorem \ref{['thm:LDP']}
  15. Case $\beta\in(0,1)$.
  16. ...and 4 more sections

Key Result

Theorem 1

Let $G_t(x)$ be non-decreasing$\forall \, t\geq0$ with $\lim_{x\to-\infty}G_t(x)=0$. Suppose that there exists $(s_t)_{t\geq 0}$ with $s_t=o(\xi_t)$ and $s_t\to 0$ as $t\to\infty$ such that, $\forall\,\lambda>0$ and some $\alpha\geq 0$, the family $(\hat{G}_t(s))_{t\geq 0}$ satisfies where $(L_t)_{t\geq 0}\in\mathcal{L}_\alpha^+$. Then, defining $x_t\coloneqq 1/s_t$,

Theorems & Definitions (39)

  • Definition
  • Definition 1
  • Remark 1
  • Theorem 1: uniform Tauberian theorem
  • Remark 2
  • Theorem 2
  • Remark 3
  • Remark 4
  • Proposition 3
  • proof
  • ...and 29 more