Sample Correlation Behavior for the Heavy Tailed General Bilinear Process

In this paper, we consider the class of general bilinear models given by X t = Z t + p X i=1 i X t?i + q X j=1 j Z t?j + m X i=1 l X j=1 b ij X t?i Z t?j ; t 2 Z; where fZ t g is an i.i.d. sequence of heavy tailed noise variables, and where Q l j=1 b 1j 6 = 0. By means of a point process analysis, we show that the sample correlation function converges in distribution to a nondegenerate random variable. Thus standard model selection and tting tools when applied to nonlinear heavy tailed models will be misleading. Also, consistency of the Hill estimator as an estimate of the tail index for this class of bilinear models is proved. 1. Introduction Currently an important topic in time series analysis is how to deal with data which exhibit features like long range dependence, nonlinearity and heavy tails. Many datasets from elds such as telecommunications, nance and economics appear to be compatible with the assumption of heavy tailed marginals. Examples include le lengths, CPU time to complete a job, call holding times, interarrival times between packets in a network and lengths of on/oo cycles. (See 9, 10, 25]) A pivotal question is how to t models to such data. In the traditional setting of stationary time series with nite variance, every purely nondeterministic process can be represented as a linear process driven by an uncorrelated input sequence. For such processes, the autocorrelation function (ACF) can be well approximated by that of a nite order ARMA(p; q) model. In particular, one can choose an autoregressive (AR) model of order p, such that the ACF's of the two models agree for lags 1; : : : ; p.. So from a second order point of view, linear models are suucient for data analysis.