Orthogonal series and limit theorems for canonical U- and V-statistics of stationary connected observations

TL;DR

This paper investigates the asymptotic distributions of canonical U- and V-statistics derived from stationary, mixing sequences, representing their limits as multilinear forms of Gaussian sequences.

Contribution

It provides a new representation of the limit distributions of degenerate U- and V-statistics for stationary sequences under mixing conditions.

Findings

Limit distributions are expressed as infinite multilinear forms of Gaussian sequences.

Results apply to arbitrary order U- and V-statistics with degenerate kernels.

The approach generalizes classical limit theorems for dependent data.

Abstract

The limit behavior is studied for the distributions of normalized U- and V-statistics of an arbitrary order with canonical (degenerate) kernels, based on samples of increasing sizes from a stationary sequence of observations satisfying classical mixing conditions. The corresponding limit distributions are represented as infinite multilinear forms of a centered Gaussian sequence with a known covariance matrix.