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Moments and tails of Lq-valued chaoses based on independent variables with log-concave tails

Rafał Meller

TL;DR

This work develops two-sided moment bounds for $L_q$-valued chaos of order two generated by independent symmetric variables with log-concave tails. By reducing to the decoupled form $S' = \sum_{i,j} a_{ij} X_i Y_j$ with $a_{ij} \in L_q$, the authors obtain a general lower bound in any Banach space and two upper bounds for the $L_q$ setting: one under a sub-Gaussian assumption and one unconditional but non-optimal. A central technical contribution is a robust machinery for bounding the expected supremum of Gaussian and exponential processes via entropy and a Main Decomposition Lemma, enabling precise two-sided moment estimates for Weibull-tailed variables with tail $\exp(-t^r)$, $r \ge 1$. The results extend the understanding of non-Gaussian chaoses in Banach spaces and provide practical moment and tail bounds with potential applications to stochastic systems and Weibull-type data.

Abstract

We derive a lower bound for moments of random chaoses of order two with coefficients in arbitrary Banach space F generated by independent symmetric random variables with logarithmically concave tails (which is probably two-sided). We also provide two upper bounds for moments of such chaoses when F = L_q. The first is true under the additional subgaussanity assumption. The second one does not require additional assumptions but is not optimal in general. Both upper bounds are sufficient for obtaining two-sided moment estimates for chaoses with values in Lq generated by Weibull random variables with shape parameter greater or equal to 1.

Moments and tails of Lq-valued chaoses based on independent variables with log-concave tails

TL;DR

This work develops two-sided moment bounds for -valued chaos of order two generated by independent symmetric variables with log-concave tails. By reducing to the decoupled form with , the authors obtain a general lower bound in any Banach space and two upper bounds for the setting: one under a sub-Gaussian assumption and one unconditional but non-optimal. A central technical contribution is a robust machinery for bounding the expected supremum of Gaussian and exponential processes via entropy and a Main Decomposition Lemma, enabling precise two-sided moment estimates for Weibull-tailed variables with tail , . The results extend the understanding of non-Gaussian chaoses in Banach spaces and provide practical moment and tail bounds with potential applications to stochastic systems and Weibull-type data.

Abstract

We derive a lower bound for moments of random chaoses of order two with coefficients in arbitrary Banach space F generated by independent symmetric random variables with logarithmically concave tails (which is probably two-sided). We also provide two upper bounds for moments of such chaoses when F = L_q. The first is true under the additional subgaussanity assumption. The second one does not require additional assumptions but is not optimal in general. Both upper bounds are sufficient for obtaining two-sided moment estimates for chaoses with values in Lq generated by Weibull random variables with shape parameter greater or equal to 1.
Paper Structure (9 sections, 48 theorems, 209 equations)

This paper contains 9 sections, 48 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Notation, convention and main results
  3. Reduction to a bound on the supremum of a certain stochastic process
  4. Expectation of suprema of a certain Gaussian process
  5. Expectation of suprema of a certain Exponential process
  6. Main Decomposition Lemma
  7. Proofs of the main results
  8. Appendix
  9. Glossary

Key Result

Proposition 2.1

Assume that $X_1,X_2,\ldots,Y_1,Y_2,\ldots$ are independent symmetric r.v.'s with LCT such that the normalization condition normalizacja holds. Let $(a_{ij})_{ij}$ belong to a Banach space $(F,\left\lVert \cdot \right\rVert)$. Then for any $p\geq 1$ we have Moreover, for any $t\geq 1$

Theorems & Definitions (94)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • Lemma 3.1
  • ...and 84 more