Stable limit theorem for \(U\)-statistic processes indexed by a random walk. (English) Zbl 1357.60025

Franke, Brice; Pène, Françoise; Wendler, Martin

Electron. Commun. Probab. 22, Paper No. 9, 12 p. (2017).

Summary: 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 \(U_n\).

MSC:

60F05	Central limit and other weak theorems
60F17	Functional limit theorems; invariance principles
60F15	Strong limit theorems
60G50	Sums of independent random variables; random walks
60K37	Processes in random environments

Keywords:

stable limit theorems; weak convergence; law of iterated logarithm; random walk; random scenery; \(U\)-statistics