Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Discrete rough paths and limit theorems

Yanghui Liu and Samy Tindel

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Abstract

In this article, we consider limit theorems for some weighted type random sums (or discrete rough integrals). We introduce a general transfer principle from limit theorems for unweighted sums to limit theorems for weighted sums via rough path techniques. As a by-product, we provide a natural explanation of the various new asymptotic behaviors in contrast with the classical unweighted random sum case. We apply our principle to derive some weighted type Breuer–Major theorems, which generalize previous results to random sums that do not have to be in a finite sum of chaos. In this context, a Breuer–Major type criterion in notion of Hermite rank is obtained. We also consider some applications to realized power variations and to Itô’s formulas in law. In the end, we study the asymptotic behavior of weighted quadratic variations for some multi-dimensional Gaussian processes.

Résumé

Dans cet article, nous étudions les théorèmes limite pour des sommes aléatoires pondérées (ou intégrales discrètes rugueuses). Nous introduisons un principe de transfert général entre les théorèmes limite pour les sommes non pondérées et pour les sommes pondérées, en utilisant des techniques de chemins rugueux. Comme conséquence, nous proposons une explication naturelle pour la diversité des nouveaux comportements asymptotiques par rapport aux cas des sommes aléatoires non pondérées. Nous appliquons notre principe pour obtenir des théorèmes de type Breuer–Major pondérés, qui généralisent des résultats précédents aux cas de sommes aléatoires qui ne sont pas dans une somme finie de chaos. Dans ce contexte, un critère de type Breuer–Major en termes de rang d’Hermite est obtenu. Nous considérons aussi des applications pour réaliser des variations de puissance et pour les formules d’Itô en loi. A cette fin, nous étudions le comportement asymptotique de variations quadratiques pondérées pour des processus Gaussiens multidimensionnels.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 3 (2020), 1730-1774.

Dates
Received: 9 August 2018
Revised: 2 July 2019
Accepted: 3 July 2019
First available in Project Euclid: 26 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1593137307

Digital Object Identifier
doi:10.1214/19-AIHP1015

Mathematical Reviews number (MathSciNet)
MR4116706

Subjects
Primary: 60B10: Convergence of probability measures 60G15: Gaussian processes 60G22: Fractional processes, including fractional Brownian motion 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Discrete rough paths Discrete rough integrals Weighted random sums Limit theorems Breuer–Major theorem

Citation

Liu, Yanghui; Tindel, Samy. Discrete rough paths and limit theorems. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 3, 1730--1774. doi:10.1214/19-AIHP1015. https://projecteuclid.org/euclid.aihp/1593137307


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