Limit theorems for von Mises statistics of a measure preserving transformation

Original · EN

For a measure preserving transformation T of a probability space (X,F,μ) we investigate almost sure and distributional convergence of random variables of the form x → 1/Cₙ ∑ᵢ₁<ₙ,...,ᵢd<ₙ f(Tⁱ¹x,...,Tⁱᵈx), n=1,2,..., where f (called the kernel) is a function from Xᵈ to and C₁, C₂,... are appropriate normalizing constants. We observe that the above random variables are well defined and belong to Lᵣ(μ) provided that the kernel is chosen from the projective tensor product Lₚ(X₁,F₁, μ₁) ⊗π...⊗π Lₚ(Xd,Fd, μd)⊂ Lₚ(μᵈ) with p=dr, r∈ [1, ∞). 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 ∑ₘ₌₁∞ λₘη²ₘ in independent standard Gaussian variables η₁, η₂,.... Our results on the distributional convergence use a T--invariant filtration as a prerequisite and are derived from uni- and multivariate martingale approximations.