Summary.
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.
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Received: 28 February 1996 / In revised form: 10 July 1996
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Giraitis, L., Taqqu, M. Limit theorems for bivariate Appell polynomials. Part I: Central limit theorems. Probab Theory Relat Fields 107, 359–381 (1997). https://doi.org/10.1007/s004400050089
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DOI: https://doi.org/10.1007/s004400050089
- Mathematics Subject Classification (1991):60F05, 62M10