The sample ACF of a simple bilinear process
Let \(X_t\) be a bilinear process defined by \[ X_t = aX_{t-1} + bX_{t-1}Z_{t-1} + Z_t, \] where \(Z_t\) are iid random variables with standard normal distribution \(\mathcal N(0,1) \) and \(a, b \) are real constants such that \(E|a + b Z_1|^\alpha = 1\) for some \(\alpha > 0.\) Then \(X_t\) has a distribution with regularly varying tails of the same index \(\alpha\). An asymptotic distribution of the sample autocorrelations of \(X_t\) is studied in three different cases, namely for \(\alpha \in (0,2),\;\alpha \in (2,4)\) and \(\alpha > 4 .\) While in the latter case the standard central limit theory for strongly mixing conditions is used, in the first two cases a point processes methodology developed recently by \textit{R. A. Davis} and \textit{T. Hsing} [Ann. Probab. 23, No. 2, 879-917 (1995; Zbl 0837.60017)] and by \textit{R. A. Davis} and \textit{T. Mikosch} [Ann. Stat. 26, No. 5, 2049-2080 (1998; Zbl 0929.62092)] is applied to obtain limiting distributions.
scientific article; zbMATH DE number 1792527