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2017 Stable limit theorem for $U$-statistic processes indexed by a random walk
Brice Franke, Françoise Pène, Martin Wendler
Electron. Commun. Probab. 22(none): 1-12 (2017). DOI: 10.1214/16-ECP4173

Abstract

Let $(S_n)_{n\in \mathbb{N} }$ be a $\mathbb{Z} $-valued random walk with increments from the domain of attraction of some $\alpha $-stable law and let $(\xi (i))_{i\in \mathbb{Z} }$ be a sequence of iid random variables. We want to investigate $U$-statistics indexed by the random walk $S_n$, that is $U_n:=\sum _{1\leq i<j\leq n}h(\xi (S_i),\xi (S_j))$ for some symmetric bivariate function $h$. We will prove the weak convergence without assumption of finite variance. Additionally, under the assumption of finite moments of order greater than two, we will establish a law of the iterated logarithm for the $U$-statistic $U_n$.

Citation

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Brice Franke. Françoise Pène. Martin Wendler. "Stable limit theorem for $U$-statistic processes indexed by a random walk." Electron. Commun. Probab. 22 1 - 12, 2017. https://doi.org/10.1214/16-ECP4173

Information

Received: 9 March 2015; Accepted: 16 March 2016; Published: 2017
First available in Project Euclid: 14 January 2017

zbMATH: 1357.60025
MathSciNet: MR3607804
Digital Object Identifier: 10.1214/16-ECP4173

Subjects:
Primary: 60F05, 60F17, 60G50, 60K37

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