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Noncentral moderate deviations for time-changed multivariate Lévy processes with linear combinations of inverse stable subordinators

Neha Gupta, Claudio Macci

TL;DR

This work extends noncentral moderate deviations to time-changed, multivariate Lévy processes where the random time changes are linear combinations of independent inverse stable subordinators. It develops two structural frameworks, cond:* and cond:**, and establishes a reference LDP with speed $t$, along with weak convergence results and noncentral MD principles with speed $1/a_t$ for scalings $(a_t)$. The authors derive explicit MD rate functions in key cases (e.g., $\nu_0<\min\{\nu_i\}$ and $\nu_0>\max\{\nu_i\}$) and provide a general MD-rate framework via Legendre–Fenchel transforms of Mittag–Leffler-type generating functions. By connecting univariate results to a multivariate setting, and employing the Gärtner–Ellis theorem, the paper delivers concrete asymptotic tools for systems whose dynamics are modulated by random time changes, with potential applications in finance, physics, and queueing theory.

Abstract

The term noncentral moderate deviations is used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between the convergence in probability to a constant (governed by a reference large deviation principle) and a weak convergence to a non-Gaussian (and non-degenerating) distribution. Some noncentral moderate deviation results in the literature concern time-changed univariate Lévy processes, where the time-changes are given by inverse stable subordinators. In this paper we present analogue results for multivariate Lévy processes; in particular the random time-changes are suitable linear combinations of independent inverse stable subordinators.

Noncentral moderate deviations for time-changed multivariate Lévy processes with linear combinations of inverse stable subordinators

TL;DR

This work extends noncentral moderate deviations to time-changed, multivariate Lévy processes where the random time changes are linear combinations of independent inverse stable subordinators. It develops two structural frameworks, cond:* and cond:**, and establishes a reference LDP with speed , along with weak convergence results and noncentral MD principles with speed for scalings . The authors derive explicit MD rate functions in key cases (e.g., and ) and provide a general MD-rate framework via Legendre–Fenchel transforms of Mittag–Leffler-type generating functions. By connecting univariate results to a multivariate setting, and employing the Gärtner–Ellis theorem, the paper delivers concrete asymptotic tools for systems whose dynamics are modulated by random time changes, with potential applications in finance, physics, and queueing theory.

Abstract

The term noncentral moderate deviations is used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between the convergence in probability to a constant (governed by a reference large deviation principle) and a weak convergence to a non-Gaussian (and non-degenerating) distribution. Some noncentral moderate deviation results in the literature concern time-changed univariate Lévy processes, where the time-changes are given by inverse stable subordinators. In this paper we present analogue results for multivariate Lévy processes; in particular the random time-changes are suitable linear combinations of independent inverse stable subordinators.

Paper Structure

This paper contains 11 sections, 9 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries on large deviations
  4. Preliminaries on Lévy processes in Assumption \ref{['claim:on-Levy-processes']}
  5. Preliminaries on inverse stable subordinators
  6. Results under Condition \ref{['cond:*']}
  7. Some cases with an explicit expression for $I_{\mathrm{MD}}(\cdot;m)$
  8. On the case $\nu_0<\min\{\nu_1,\ldots,\nu_h\}$
  9. On the case $\nu_0>\max\{\nu_1,\ldots,\nu_h\}$ with $m\neq 0$
  10. Results under Condition \ref{['cond:**']}
  11. Acknowledgements.

Key Result

Theorem 2.1

Assume that, for all $\theta\in\mathbb{R}^b$, there exists as an extended real number (here $\langle\cdot,\cdot\rangle$ is the inner product in $\mathbb{R}^b$). Assume that $\theta=0\in\mathbb{R}^b$ belongs to the interior of the set $\mathcal{D}(\Lambda):=\{\theta\in\mathbb{R}^b:\Lambda(\theta)<\infty\}$; moreover assume that the function $\Lambda$ is esse (i.e. $\Lambda^*$ is the Legendre-Fench

Theorems & Definitions (21)

  • Remark 1.1
  • Remark 1.2
  • Theorem 2.1: Gärtner Ellis Theorem
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.1
  • ...and 11 more