Research has concentrated on probability modeling, with emphasis on extreme value theory and modeling of phenomena requiring heavy tails. These are elegant and mathematically fascinating theories with an enormous variety of applications. A mathematical theme has been the usefulness of point processes for the study of weak convergence phenomena connected with heavy tails. An analytic property called regular variation, which specifies the weight of a distribution tail, has a probabilistic analogue in the weak convergence of induced point processes to a Poisson limit.

Work continues on understanding how to fit heavy tailed models to data and in understanding the influence of heavy tails and induced long range dependence on data networks. Recent effort has focussed on the inability of standard data network models to adequately explain observed phenomena in data traces. Fluid queues have frequently been used as models for data networks and we focussed on models where a server works off the load at constant rate r. The server is fed by sources or nodes. A variety of assumptions are possible, such as the server being fed by (a) a single on-off renewal process, (b) k such on-off models, (c) an infinite number of sources. In models (a) and (b) sources turn on at renewal time points and in (c) sources initiate connections at Poisson times. Transmissions last for iid periods governed by a heavy--tailed distribution of session lengths. The heavy tails induce long range dependence in the system and result in performance deterioration.

Mikosch, Resnick, Rootzen and Stegeman (1999) study both the the infinite source Poisson model and the superposition of renewal on/off inputs model and show that the cumulative input can be approximated by either Levy stable motion or fractional Brownian motion depending on the interaction of connection rates and tail properties of the connection or on-period length distribution. An interesting project with the Goteborg Stochastic Center in Sweden examined available internet data sets in order to analyze in what ways the infinite source Poisson model fit the data and in what ways it was deficient. Available statistical techniques were surveyed and applied. We reviewed estimators for the Hurst and Holder parameters and the tail indices of marginal distributions. General conclusions were that the model, although offering an explanation of induced long range dependence, did not fit the data well and that one source of error was that transmission rates are never constant as supposed in the model. This conclusion led to studying a model where transmission rates and transmission times of files are (possibly) dependent heavy tailed random variables. This generalization provided more reality but does not explain observed multifractality on very fine time scales. So a further generalization is under consideration which allows modeling transmission via randomly time varying schedules which exhibit multifractality on fine time scales but a form of self-similarity at coarse time scales.

Recent projects in extreme value theory have centered on understanding asymptotic independence and considering whether asymptotic independence can be detected statistically. In studying such things as exchange rate returns between currencies relative to, say the US dollar, very different behavior is noted for bivariate behavior depending on which two country's are compared. Detecting asymptotic dependence is one way to explain different behavior which is more refined than the usual dependence measures such as correlation. We have studied a sub-model of asymptotic independence called ''hidden regular variation'' (building on ideas of Tawn, Coles, Heffernan and others) which allows for investigation of statistical procedures for detection.

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