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Exponential tilting of subweibull distributions

F. William Townes

TL;DR

This work unifies subexponential and subgaussian tails via the $q$-subweibull framework and develops two equivalent characterizations, including tail bounds, moment growth, and Orlicz-type norms. It formalizes radius-of-convergence $R_q$ as a key descriptor and analyzes how exponential tilting affects tail behavior, proving preservation of $q$-subweibull tails (with the same $R_q$) for nonnegative variables when $q>1$, while showing heavier-tailed cases can become lighter under tilting. A notable result is that the Poisson distribution is strictly subexponential and strictly $q$-subweibull only for $q\le 1$, highlighting nuanced thresholds in tail heaviness. The findings have implications for concentration inequalities, stochastic modeling, and sampling methods that rely on tilting, providing precise conditions under which tail properties are maintained or transformed under exponential tilting.

Abstract

The class of subweibull distributions has recently been shown to generalize the important properties of subexponential and subgaussian random variables. We describe alternative characterizations of subweibull distributions and detail the conditions under which their tail behavior is preserved after exponential tilting.

Exponential tilting of subweibull distributions

TL;DR

This work unifies subexponential and subgaussian tails via the -subweibull framework and develops two equivalent characterizations, including tail bounds, moment growth, and Orlicz-type norms. It formalizes radius-of-convergence as a key descriptor and analyzes how exponential tilting affects tail behavior, proving preservation of -subweibull tails (with the same ) for nonnegative variables when , while showing heavier-tailed cases can become lighter under tilting. A notable result is that the Poisson distribution is strictly subexponential and strictly -subweibull only for , highlighting nuanced thresholds in tail heaviness. The findings have implications for concentration inequalities, stochastic modeling, and sampling methods that rely on tilting, providing precise conditions under which tail properties are maintained or transformed under exponential tilting.

Abstract

The class of subweibull distributions has recently been shown to generalize the important properties of subexponential and subgaussian random variables. We describe alternative characterizations of subweibull distributions and detail the conditions under which their tail behavior is preserved after exponential tilting.
Paper Structure (6 sections, 11 theorems, 41 equations)

This paper contains 6 sections, 11 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Laplace transforms
  3. Subweibull random variables
  4. Subweibull properties of the Poisson distribution
  5. Exponential tilting
  6. Discussion

Key Result

Lemma 2.0.1

If the Laplace transforms of random variables $X$ and $Y$ satisfy $\mathcal{L}_X(t) = \mathcal{L}_Y(t)$ for all $t$ in any nonempty open interval $(a,b)\subset \mathbb{R}$, not necessarily containing zero, then $X\overset{d}{=} Y$.

Theorems & Definitions (24)

  • Definition 2.1
  • Lemma 2.0.1
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.0.1
  • proof
  • Proposition 3.1
  • Corollary 3.0.1
  • proof
  • Corollary 3.0.2
  • ...and 14 more