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Limit theorems for bivariate Appell polynomials. I: Central limit theorems. (English) Zbl 0873.60007

From the Summary: Consider the stationary linear process \(X_t=\sum_{u=-\infty}^{\infty} a(t-u) \xi_u\), \(t\in {\mathcal Z}\), 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 nonlinear 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 behavior holds if \(b\) and \(a\) are not too badly behaved. However, the good behaviour of one kernel may compensate for the bad behaviour of the other. The conditions are formulated in the spectral domain.

MSC:

60F05 Central limit and other weak theorems
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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