We have recently been investigating the role of long-range dependence and heavy tails in the stochastic phenomena that drive the evolution of systems. Some general results in this area will be presented in this talk. There are two major themes in this talk. One is a discussion of how stochastic approximation theory works in the presence of long-range dependence and heavy tails in the noise. A development parallel to the traditional theory in the usual short-range dependent case will be outlined. The main result of this work (joint work with Vivek Borkar) has the potential to be applied in many of the application scenarios where stochastic approximation algorithms are used, when long-range dependence and/or heavy tailed behavior is a concern. The second main theme considered is when a function of a long-range dependent process is itself long-range dependent. A general theorem along these lines will be presented and discussed, together with some illustrative examples (to data compression and financial time series, and possibly others, time permitting) to suggest that such a result has broad applicability (joint work with Barlas Oguz). |
