## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 22 (2017), paper no. 9, 12 pp.

### Stable limit theorem for $U$-statistic processes indexed by a random walk

Brice Franke, Françoise Pène, and Martin Wendler

#### 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 $U_n$.

#### Article information

**Source**

Electron. Commun. Probab., Volume 22 (2017), paper no. 9, 12 pp.

**Dates**

Received: 9 March 2015

Accepted: 16 March 2016

First available in Project Euclid: 14 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1484363136

**Digital Object Identifier**

doi:10.1214/16-ECP4173

**Mathematical Reviews number (MathSciNet)**

MR3607804

**Zentralblatt MATH identifier**

1357.60025

**Subjects**

Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60G50: Sums of independent random variables; random walks 60K37: Processes in random environments

**Keywords**

random walk random scenery $U$-statistics stable limits law of the iterated logarithm

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Franke, Brice; Pène, Françoise; Wendler, Martin. Stable limit theorem for $U$-statistic processes indexed by a random walk. Electron. Commun. Probab. 22 (2017), paper no. 9, 12 pp. doi:10.1214/16-ECP4173. https://projecteuclid.org/euclid.ecp/1484363136