Limit theorems for bivariate Appell polynomials. Part I: Central limit theorems

Abstract

Consider the stationary linear process $X_t=\sum_{u=-\infty}^\infty a(t-u)\xi_u$ , $t\in {\bf Z}$ , where $\{ \xi_u\}$ is an i.i.d. finite variance sequence. The spectral density of $\{ X_t\}$ may diverge at the origin (long-range dependence) or at any other frequency. Consider now the quadratic form $Q_N=\sum_{t,s=1}^N b(t-s)P_{m,n} (X_t,X_s)$ , where $P_{m,n}(X_t,X_s)$ denotes a non-linear function (Appell polynomial). We provide general conditions on the kernels $b$ and $a$ for $N^{-1/2}Q_N$ to converge to a Gaussian distribution. We show that this convergence holds if $b$ and $a$ are not too badly behaved. However, the good behavior of one kernel may compensate for the bad behavior of the other. The conditions are formulated in the spectral domain.