Limit theorems for von Mises statistics of a measure preserving transformation

Abstract

For a measure preserving transformation $$T$$ of a probability space $$(X,\mathcal{F },\mu )$$ and some $$d \ge 1$$ we investigate almost sure and distributional convergence of random variables of the form $$\begin{aligned} x \rightarrow \frac{1}{C_n} \sum _{0\le i_1,\ldots ,\,i_d<n} f(T^{i_1}x,\ldots ,T^{i_d}x),\ n=1,2, \ldots , \end{aligned}$$ where $$C_1, C_2,\ldots $$ are normalizing constants and the kernel $$f$$ belongs to an appropriate subspace in some $$L_p(X^d\!,\, \mathcal{F }^{\otimes d}\!,\,\mu ^d)$$ . We establish a form of the individual ergodic theorem for such sequences. Using a filtration compatible with $$T$$ and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and $$d=2$$ , we also show that the convergence holds in distribution towards a quadratic form $$\sum _{m=1}^{\infty } \lambda _m\eta ^2_m$$ in independent standard Gaussian variables $$\eta _1, \eta _2, \ldots $$ .