Multifractal Analysis of a Class of Additive Processes with Correlated Non-Stationary Increments

Abstract

We consider a family of stochastic processes built from infinite sums of independent positive random functions on $R_+$. Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on $R_+$. The motivation for studying these processes stems from the fact that they constitute simplified models for TCP traffic on the Internet. Such processes bear some analogy with Lévy processes, but they are more complex in the sense that their increments are neither stationary nor independent. Nevertheless, we show that their multifractal behavior is very much the same as that of certain Lévy processes. More precisely, we compute the Hausdorff multifractal spectrum of our processes, and find that it shares the shape of the spectrum of a typical Lévy process. This result yields a theoretical basis to the empirical discovery of the multifractal nature of TCP traffic.