Limit theory for bilinear processes with heavy-tailed noise

Abstract

We consider a simple stationary bilinear model $X_t = cX_{t-1} Z_{t-1} + Z_t, t = 0, \pm 1, \pm 2, \dots,$ generated by heavy-tailed noise variables ${Z_t}$. A complete analysis of weak limit behavior is given by means of a point process analysis. A striking feature of this analysis is that the sample correlation converges in distribution to a nondegenerate limit. A warning is sounded about trying to detect nonlinearities in heavy-tailed models by means of the sample correlation function.