Multifractal Products of Stochastic Processes: a Preview

Motivated by the need for multifractal processes with stationary in crements we introduce a construction of random multifractal measures based on iterative multiplication of stationary stochastic processes We establish conditions for the L convergence and non degeneracy of the limit process in a general setting Proceeding then to multiply ing piecewise constant processes we proof continuity of the limit and show some other interesting properties Introduction This study is strongly motivated by the seek of new models for teletra c In various recent papers see e g RLV LVR MN FGW it has been demonstrated that teletra c exhibits multifractal properties There are many ways to construct random multifractal measures varying from the simple binomial measures to measures generated by branching processes see e g Man Man Fal AP Pat RCRB In teletra c model ing we would like to have in addition to simplicity of the construction also stationarity of the increments Unfortunately most of the multifractal models introduced so far lack this property Although Ja ard has shown that L evy processes are multifractal Jaf but unfortunately from the point of teletra c modeling increments of a L evy process are in addition to stationary also independent Moreover L evy processes have a linear mul tifractal spectrum while real data tra c exhibits strictly concave spectra RLV LVR MN RCRB VTT Information Technology P O Box FIN VTT Finland E mail petteri mannersalo vtt VTT Information Technology P O Box FIN VTT Finland E mail ilkka norros vtt ECE Dept Rice University P O Box Houston TX USA E mail riedi rice edu In its simplest form our model is based on the multiplication of inde pendent rescaled stochastic processes i d b which are piecewise constant It is instructive to compare it to a Fourier decomposition where one represents or constructs a process by superposition of oscillations sin it In multiplying rather than adding rescaled versions of a mother process we obtain a process with novel properties which are best understood not in an additive analysis but in a multiplicative one Moreover processes emerging from multiplicative construction schemes have positive increments and exhibit typically a spiky appearance The so called multifractal analysis describes the local structure of a process in terms of scaling exponents accounting for being adapted to the multiplicative structure With our scheme we generalize the construction of the binomial cascade in randomizing it in a natural and stationary way As with the cascades an in nite product of random processes will typically be zero and one has to take a distributional limit rather than pointwise limit In more simple words a multiplier i t should not be evaluated in points but should be seen as redistributing or re partitioning mass Again in other words i t can be thought of as a local change in the arrival rate where one is interested actually in the integrated total load process Consequently we set An t Z t n s ds Z t n Y