Heavy tail modeling in time series and telecommunications

The work presented in this thesis concentrates on modeling with heavy tailed random variables, motivated by applications in telecommunications, economics and finance.; Chapter 1 discusses heavy tailed time series. A heavy tailed time series which can be represented as an infinite moving average (MA($\infty$)) has the property that the sample auto correlation function at lag h converges to a constant $rh$. However, this property of MA($\infty$) is an exception among heavy tailed models. The main result of the first chapter, shows that for a large class of general bilinear models, the sample ACF at lag h converges in distribution to a nondegenerate random variable depending on h. Thus, standard tools for selecting and fitting ARMA models will be misleading when applied to these nonlinear heavy tailed models.; The fact that the convergence of the sample ACF to a constant is not true for general heavy tailed models, suggests a test for (non)linearity, and in Chapter 2 a test, based on subsample stability of the sample ACF, is developed. The test is applied to several real network communications datasets, and simulated bilinear datasets. The test results indicate strongly, that a linear model is inappropriate for the network communication data sets.; Chapter 3 discusses heavy tail modeling in telecommunications, and focusses on weak convergence of a high-speed network traffic model in Skorohod space, equipped with the Skorohod $M1$ topology. The physical model is a high speed network with finite bandwidth fed by a large number of traffic sources. Each source transmits at a possibly variable rate for a period of time equal to the session duration, then stays silent for a while, and the cycle repeats. The distribution of the session duration is assumed to be heavy tailed, with infinite variance. We consider the total volume of traffic injected into the network between time 0 and Tt. Scaled appropriately, the finite dimensional distributions of the cummulative traffic converge to these of a skewed Lévy stable motion as $T\to \infty$. Showing convergence in Skorohod space allows us to prove a heavy traffic theorem for a single server fluid queue.