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

Antonella Iuliano, Claudio Macci, Alessandra Meoli

TL;DR

The paper addresses noncentral moderate deviations for time-changed Lévy processes driven by inverse stable subordinators, extending prior results to general Lévy drivers and to differences of independent subordinators. It develops one- and two-time-change frameworks, deriving large deviation principles with explicit rate functions and corresponding noncentral moderate deviation results, leveraging Mittag-Leffler asymptotics and Gärtner–Ellis theory. The work also introduces generalizations to broader time-changes and provides comparisons between rate functions, illustrating the theory with tempered stable subordinators. Collectively, the results offer a flexible, analytically tractable framework for noncentral MD behavior in fractional/time-changed stochastic processes, with potential applications in areas requiring precise small-probability asymptotics. The analysis yields explicit MD rate forms and highlights when MD behavior is non-Gaussian and how it depends on drift and variation properties of the underlying Lévy process.

Abstract

In this paper we present some extensions of recent noncentral moderate deviation results in the literature. In the first part we generalize the results in \cite{BeghinMacciSPL2022} by considering a general Lévy process $\{S(t):t\geq 0\}$ instead of a compound Poisson process. In the second part we assume that $\{S(t):t\geq 0\}$ has bounded variation and is not a subordinator; thus $\{S(t):t\geq 0\}$ can be seen as the difference of two independent non-null subordinators. In this way we generalize the results in \cite{LeeMacci} for Skellam processes.

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

TL;DR

The paper addresses noncentral moderate deviations for time-changed Lévy processes driven by inverse stable subordinators, extending prior results to general Lévy drivers and to differences of independent subordinators. It develops one- and two-time-change frameworks, deriving large deviation principles with explicit rate functions and corresponding noncentral moderate deviation results, leveraging Mittag-Leffler asymptotics and Gärtner–Ellis theory. The work also introduces generalizations to broader time-changes and provides comparisons between rate functions, illustrating the theory with tempered stable subordinators. Collectively, the results offer a flexible, analytically tractable framework for noncentral MD behavior in fractional/time-changed stochastic processes, with potential applications in areas requiring precise small-probability asymptotics. The analysis yields explicit MD rate forms and highlights when MD behavior is non-Gaussian and how it depends on drift and variation properties of the underlying Lévy process.

Abstract

In this paper we present some extensions of recent noncentral moderate deviation results in the literature. In the first part we generalize the results in \cite{BeghinMacciSPL2022} by considering a general Lévy process instead of a compound Poisson process. In the second part we assume that has bounded variation and is not a subordinator; thus can be seen as the difference of two independent non-null subordinators. In this way we generalize the results in \cite{LeeMacci} for Skellam processes.
Paper Structure (11 sections, 13 theorems, 73 equations, 4 figures)

This paper contains 11 sections, 13 theorems, 73 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries on large deviations
  4. Preliminaries on the inverse of a stable subordinator
  5. Results with only one random time-change
  6. Results with two independent random time-changes
  7. A discussion on possible generalizations
  8. Comparisons between rate functions
  9. An example with independent tempered stable subordinators
  10. Acknowledgements.
  11. Funding.

Key Result

Theorem 2.1

Assume that, for all $\theta\in\mathbb{R}$, there exists as an extended real number; moreover assume that the origin $\theta=0$ belongs to the interior of the set $\mathcal{D}(\Lambda):=\{\theta\in\mathbb{R}:\Lambda(\theta)<\infty\}$. Furthermore let $\Lambda^*$ be the Legendre-Fenchel transform of $\Lambda$, i.e. the function defined by Then, if $\Lambda$ is essentially smooth and lower semi-co

Figures (4)

  • Figure 1: The function $\Lambda_{\nu,S}$ in Remark \ref{['rem:es-not-guaranteed']} for $\theta\leq r=1$. Numerical values: $\nu=0.5$, $\beta=0.25$; $h=0.5$ on the left, and $h=3$ on the right.
  • Figure 2: Rate functions $I_{\mathrm{LD}}=I_{\mathrm{LD},\nu}$ (on the left) and $J_{\mathrm{LD}}=J_{\mathrm{LD},\nu}$ (on the right) for the processes in Example \ref{['ex:diff-TSS']} and different values of $\nu$ ($\nu=0.1,0.5,0.9$). Numerical values of the other parameters: $r_1=1$, $r_2=2$, $\beta=0.5$.
  • Figure 3: Rate functions $I_{\mathrm{LD}}$ and $J_{\mathrm{LD}}$ for the processes in Example \ref{['ex:diff-TSS']} and different values of $\beta$ ($\beta=0.3,0.5,0.7$ from left to right). Numerical values of the other parameters: $r_1=1$, $r_2=2$, $\nu=0.5$.
  • Figure 4: Rate functions $I_{\mathrm{LD}}$ and $J_{\mathrm{LD}}$ for the processes in Example \ref{['ex:diff-TSS']} and different values of $r_2$ ($r_2=5,10,50$ from left to right). Numerical values of the other parameters: $r_1=1$, $\nu=0.5$, $\beta=0.5$.

Theorems & Definitions (29)

  • Theorem 2.1: Gärtner Ellis Theorem
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.2
  • Remark 3.3
  • ...and 19 more