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Sample-Path Large Deviations for Lévy Processes and Random Walks with Lognormal Increments

Zhe Su, Chang-Han Rhee

Abstract

The large deviations theory for heavy-tailed processes has seen significant advances in the recent past. In particular, Rhee et al. (2019) and Bazhba et al. (2020) established large deviation asymptotics at the sample-path level for Lévy processes and random walks with regularly varying and (heavy-tailed) Weibull-type increments. This leaves the lognormal case -- one of the three most prominent classes of heavy-tailed distributions, alongside regular variation and Weibull -- open. This article establishes the \emph{extended large deviation principle} (extended LDP) at the sample-path level for one-dimensional Lévy processes and random walks with lognormal-type increments. Building on these results, we also establish the extended LDPs for multi-dimensional processes with independent coordinates. We demonstrate the sharpness of these results by constructing counterexamples, thereby proving that our results cannot be strengthened to a standard LDP under $J_1$ topology and $M_1'$ topology.

Sample-Path Large Deviations for Lévy Processes and Random Walks with Lognormal Increments

Abstract

The large deviations theory for heavy-tailed processes has seen significant advances in the recent past. In particular, Rhee et al. (2019) and Bazhba et al. (2020) established large deviation asymptotics at the sample-path level for Lévy processes and random walks with regularly varying and (heavy-tailed) Weibull-type increments. This leaves the lognormal case -- one of the three most prominent classes of heavy-tailed distributions, alongside regular variation and Weibull -- open. This article establishes the \emph{extended large deviation principle} (extended LDP) at the sample-path level for one-dimensional Lévy processes and random walks with lognormal-type increments. Building on these results, we also establish the extended LDPs for multi-dimensional processes with independent coordinates. We demonstrate the sharpness of these results by constructing counterexamples, thereby proving that our results cannot be strengthened to a standard LDP under topology and topology.

Paper Structure

This paper contains 19 sections, 32 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Extended LDP for Lévy processes
  5. Extended LDP for random walks
  6. Extended LDP for multi-dimensional processes
  7. Nonexistence of standard LDPs
  8. Example: boundary crossing with regulated jumps
  9. Proofs
  10. Proofs for Section \ref{['section:notation-definitions']}
  11. Proofs for Section \ref{['section:extended-LDP-levy-J_1']}
  12. Proof of Proposition \ref{['theorem:ex-ldp-kjumpprocess']}
  13. Proof of Lemma \ref{['lemma:checkjnk-asymp-neg']}
  14. Proofs for Section \ref{['section:randomwalk-ldp-J1']}
  15. Proofs for Section \ref{['section:extended-ldp-j1-multidimension']}
  16. ...and 4 more sections

Key Result

Lemma 2.1

Suppose that $\xi,\zeta \in \mathbb{{D}}$. Then for any $(u,r) \in \Gamma(\xi)$,

Theorems & Definitions (60)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.1
  • Lemma 2.3
  • Proposition 2.1: Proposition 2.1 of MR4493394
  • Corollary 2.1: Corollary 2.1 of MR4493394
  • Remark 1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 50 more