• Bernoulli
  • Volume 23, Number 1 (2017), 329-378.

Convergence of $U$-statistics indexed by a random walk to stochastic integrals of a Lévy sheet

Brice Franke, Françoise Pène, and Martin Wendler

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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 $Z$.

Article information

Bernoulli Volume 23, Number 1 (2017), 329-378.

Received: August 2014
Revised: June 2015
First available in Project Euclid: 27 September 2016

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Lévy sheet random scenery random walk stable limits $U$-statistics


Franke, Brice; Pène, Françoise; Wendler, Martin. Convergence of $U$-statistics indexed by a random walk to stochastic integrals of a Lévy sheet. Bernoulli 23 (2017), no. 1, 329--378. doi:10.3150/15-BEJ745.

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