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Some vector-valued examples of noncentral moderate deviation results

Claudio Macci, Barbara Pacchiarotti

TL;DR

Addresses vector-valued noncentral moderate deviations, extending the univariate theory to $\mathbb{R}^h$-valued random variables. It develops several multivariate constructions (time-changed multivariate Lévy processes, Poisson-type limits, and continuous mappings) and proves corresponding LDPs and weak convergence results. The main contributions are explicit rate functions $I_{\mathrm{LD}}$ and noncentral rate functions $I_{\mathrm{MD}}$, together with contraction-based extensions to logistic-normal and skew-normal limits. This work broadens the noncentral moderate deviation framework to multivariate settings with practical implications for dependent or heavy-tailed systems.

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. Several examples can be found in the literature, mainly for real-valued random variables (see, e.g.,~\cite{GiulianoMacci} and the references cited therein). In this paper we present some examples with vector-valued random variables.

Some vector-valued examples of noncentral moderate deviation results

TL;DR

Addresses vector-valued noncentral moderate deviations, extending the univariate theory to -valued random variables. It develops several multivariate constructions (time-changed multivariate Lévy processes, Poisson-type limits, and continuous mappings) and proves corresponding LDPs and weak convergence results. The main contributions are explicit rate functions and noncentral rate functions , together with contraction-based extensions to logistic-normal and skew-normal limits. This work broadens the noncentral moderate deviation framework to multivariate settings with practical implications for dependent or heavy-tailed systems.

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. Several examples can be found in the literature, mainly for real-valued random variables (see, e.g.,~\cite{GiulianoMacci} and the references cited therein). In this paper we present some examples with vector-valued random variables.

Paper Structure

This paper contains 6 sections, 13 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on large (and moderate) deviations
  3. Multivariate Lévy processes time-changed with an inverse stable subordinator
  4. Multivariate Lévy processes time-changed with a subordinator
  5. On some examples with a compound Poisson weak limits
  6. On some examples obtained by applying continuous mappings

Key Result

Proposition 3.1

Assume that $f_\nu\circ\kappa_S$ is an essentially smooth function. Then $\left\{\frac{S(L_\nu(t))}{t}:t>0\right\}$ satisfies the LDP with speed $t$ and good rate function $I_{\mathrm{LD}}$ defined by

Theorems & Definitions (38)

  • Remark 2.1
  • Proposition 3.1: Reference LDP
  • proof
  • Proposition 3.2: Weak convergence
  • proof
  • Proposition 3.3: Noncentral moderate deviations
  • proof
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.4
  • ...and 28 more