Limit theorems for von Mises statistics of a measure preserving transformation

Abstract

For a measure preserving transformation $T$ of a probability space $(X,\mathcal{F},\mu )$ we investigate almost sure and distributional convergence of random variables of the form $$x\to \frac{1}{C_n}\sum _{i_1<n,...,i_d<n}f(T^{i_1}x,...,T^{i_d}x),n=1,2,...,$$ where $f$ (called the \emph{kernel}) is a function from $X^d$ to $\mathbb{R}$ and $C_1,C_2,...$ are appropriate normalizing constants. We observe that the above random variables are well defined and belong to $L_r(\mu )$ provided that the kernel is chosen from the projective tensor product $$L_p(X_1,{\mathcal{F}}_1,{\mu }_1){\otimes }_{\pi }...{\otimes }_{\pi }{L}_p(X_d,{\mathcal{F}}_d,{\mu }_d)\subset L_p({\mu }^d)$$ with $p=dr,r\in [1,\infty ).$ We establish a form of the individual ergodic theorem for such sequences. Next, we give a martingale approximation argument to derive a central limit theorem in the non-degenerate case (in the sense of the classical Hoeffding's decomposition). Furthermore, for $d=2$ and a wide class of canonical kernels $f$ we also show that the convergence holds in distribution towards a quadratic form ${\sum }_{m=1}^{\infty }{\lambda }_m{\eta }_m^2$ in independent standard Gaussian variables ${\eta }_1,{\eta }_2,...$. Our results on the distributional convergence use a $T$--\,invariant filtration as a prerequisite and are derived from uni- and multivariate martingale approximations.

References in corpus (2)

• Degenerate U- and V-statistics under weak dependence: Asymptotic theory and bootstrap consistency
• Stochastic integrals and asymptotic analysis of canonical von Mises statistics based on dependent observations

Cited by in corpus (1)

• The Lindeberg theorem for Gibbs-Markov dynamics