Normal Approximation for $U$- and $V$-statistics of a Stationary Absolutely Regular Sequence

Abstract

Let $(X_{n,t})_{t=1}^{\infty}$ be a stationary absolutely regular sequence of real random variables with the distribution dependent on the number~$n$. The paper presents sufficient conditions for the asymptotic normality (for $n\to\infty$ and common centering and normalization) of the distribution of the nonhomogeneous $U$-statistic of order $r$ which is given on the sequence $X_{n,1},\ldots,X_{n,n}$ with a kernel also dependent on $n$. The same results for $V$-statistics also hold. To analyze sums of dependent random variables with rare strong dependencies, the proof uses the approach that was proposed by S.~Janson in 1988 and upgraded by V.~Mikhailov in 1991 and M.~Tikhomirova and V.~Chistyakov in 2015.