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Generalized law of iterated logarithm for the Lorentz gas with infinite horizon

Péter Bálint, Dalia Terhesiu

TL;DR

The paper develops a generalized almost sure invariance principle and a generalized law of the iterated logarithm for a class of weakly dependent, heavy-tailed processes in the non-standard domain of attraction of the normal law, with the infinite-horizon Lorentz gas as the primary example. By constructing an abstract dynamical-systems framework and verifying a suite of moment and dependence conditions (H1)-(H3) and (HL1)-(HL3), the authors obtain a sharp Gaussian-approximation (ASIP) for partial sums with an explicit slowly varying normalization c_n and a corresponding limsup constant a. The core technical achievement is a Gouëzel-type coupling for truncated block sums, complemented by a hierarchy of truncations and detailed moment bounds; this yields a generalized LIL that mirrors Einmahl’s i.i.d. results but for the dependent Lorentz-gas setting. The results establish that the Lorentz gas with infinite horizon exhibits precise almost-sure asymptotics for sums along the collision process, providing a rigorous bridge between non-standard diffusion in deterministic dynamical systems and probabilistic limit theory with heavy-tailed tails.

Abstract

We obtain a generalized law of the iterated logarithm for a class of dependent processes with superdiffusive behaviour. Our results apply in particular to the Lorentz gas with infinite horizon.

Generalized law of iterated logarithm for the Lorentz gas with infinite horizon

TL;DR

The paper develops a generalized almost sure invariance principle and a generalized law of the iterated logarithm for a class of weakly dependent, heavy-tailed processes in the non-standard domain of attraction of the normal law, with the infinite-horizon Lorentz gas as the primary example. By constructing an abstract dynamical-systems framework and verifying a suite of moment and dependence conditions (H1)-(H3) and (HL1)-(HL3), the authors obtain a sharp Gaussian-approximation (ASIP) for partial sums with an explicit slowly varying normalization c_n and a corresponding limsup constant a. The core technical achievement is a Gouëzel-type coupling for truncated block sums, complemented by a hierarchy of truncations and detailed moment bounds; this yields a generalized LIL that mirrors Einmahl’s i.i.d. results but for the dependent Lorentz-gas setting. The results establish that the Lorentz gas with infinite horizon exhibits precise almost-sure asymptotics for sums along the collision process, providing a rigorous bridge between non-standard diffusion in deterministic dynamical systems and probabilistic limit theory with heavy-tailed tails.

Abstract

We obtain a generalized law of the iterated logarithm for a class of dependent processes with superdiffusive behaviour. Our results apply in particular to the Lorentz gas with infinite horizon.
Paper Structure (23 sections, 19 theorems, 125 equations)

This paper contains 23 sections, 19 theorems, 125 equations.

Table of Contents

  1. Introduction and main results
  2. The infinite horizon Lorentz gas
  3. Classical ASIP and LIL
  4. A certain ASIP and generalized LIL for independent sequences
  5. Main result: generalized ASIP and LIL for the Lorentz gas map with infinite horizon
  6. Main ingredients for the proof of Theorem \ref{['thm-LG']}
  7. Set up
  8. Main propositions used in the proof of Theorem \ref{['thm-LG']}
  9. Notations.
  10. Proof of Theorem \ref{['thm-LG']}
  11. Proof of Proposition \ref{['thm-main']}
  12. Proof of Proposition \ref{['prop:cr']}
  13. Proofs of the lemmas used in the proof of Proposition \ref{['thm-main']}
  14. Proof of Lemma \ref{['prop:i.i.d.app']}
  15. Proof of Lemma \ref{['lemma:Gaus']}
  16. ...and 8 more sections

Key Result

Theorem 1.5

Ein07, Ein09. Let $(Y_j)_{ j\ge 0}$ be an i.i.d sequence of $\mathbb{R}^d$ random variables, regularly varying with index $-2$, and satisfying eq:law1. Let $(c_n)_{n \geq 1}$ be a sequence satisfying (H0). Then, there exists a probability space $(\Omega_1,\mathcal{F}_1, \mathbb{P}_1)$ and two sequen and so that, almost surely, as $n\to\infty$, where the matrix $\Gamma_n$ is given by $\Gamma_n^2=c

Theorems & Definitions (27)

  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 17 more