Limit theory for bilinear processes with heavy-tailed noise. (English) Zbl 0879.60053

Davis, Richard A.; Resnick, Sidney I.

Ann. Appl. Probab. 6, No. 4, 1191-1210 (1996).

The authors consider a simple first-order bilinear time series \(\{X_t\}\) defined as a stationary solution to the equations \(X_t=cX_{t-1}Z_{t-1}+ Z_t\), where \(\{Z_t\}\) is an i.i.d. sequence of random variables with regularly varying tail probabilities. Using the weak convergence techniques based on the point process method, they give a complete analysis of weak limit behaviour of \(\{X_t\}\). Some corollaries of the obtained convergence results, with emphasis on the limiting behaviour of the extremes, partial sums and sample correlations, are presented. Note that, unlike the linear process case, the sample correlations of the bilinear process converge in distribution to nondegenerate limit random variables.

Reviewer: W.Dziubdziela (Czȩstochowa)

MSC:

60G70	Extreme value theory; extremal stochastic processes
60E07	Infinitely divisible distributions; stable distributions
60F17	Functional limit theorems; invariance principles
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)

Keywords:

extreme values; point processes; bilinear and nonlinear processes; sample correlation

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