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February 2017 Convergence of $U$-statistics indexed by a random walk to stochastic integrals of a Lévy sheet
Brice Franke, Françoise Pène, Martin Wendler
Bernoulli 23(1): 329-378 (February 2017). DOI: 10.3150/15-BEJ745

Abstract

A $U$-statistic indexed by a $\mathbb{Z}^{d_{0}}$-random walk $(S_{n})_{n}$ is a process $U_{n}:=\sum_{i,j=1}^{n}h(\xi_{S_{i}},\xi_{S_{j}})$ where $h$ is some real-valued function and $(\xi_{k})_{k}$ is a sequence of i.i.d. random variables, which are independent of the walk. Concerning the walk, we assume either that it is transient or that its increments are in the normal domain of attraction of a strictly stable distribution of exponent $\alpha\in[d_{0},2]$. We further assume that the distribution of $h(\xi_{1},\xi_{2})$ belongs to the normal domain of attraction of a strictly stable distribution of exponent $\beta\in(0,2)$. For a suitable renormalization $(a_{n})_{n}$ we establish the convergence in distribution of the sequence of processes $(U_{\lfloor nt\rfloor}/a_{n})_{t};n\in\mathbb{N}$ to some suitable observable of a Lévy sheet $(Z_{s,t})_{s,t}$. The limit process is the diagonal process $(Z_{t,t})_{t}$ when $\alpha=d_{0}\in\{1,2\}$ or when the underlying walk is transient for arbitrary $d_{0}\ge1$. When $\alpha>d_{0}=1$, the limit process is some stochastic integral with respect to $Z$.

Citation

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Brice Franke. Françoise Pène. Martin Wendler. "Convergence of $U$-statistics indexed by a random walk to stochastic integrals of a Lévy sheet." Bernoulli 23 (1) 329 - 378, February 2017. https://doi.org/10.3150/15-BEJ745

Information

Received: 1 August 2014; Revised: 1 June 2015; Published: February 2017
First available in Project Euclid: 27 September 2016

zbMATH: 1367.60047
MathSciNet: MR3556775
Digital Object Identifier: 10.3150/15-BEJ745

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

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Vol.23 • No. 1 • February 2017
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