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On generalized multiplicative cascades

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  • Liu, Quansheng

Abstract

We consider a generalized Mandelbrot's martingale {Yn} and the associated Mandelbrot's measure [mu][omega] on marked trees. If the limit variable Z=lim Yn is not degenerate, we study the asymptotic behavior at infinity of its distribution; in the contrary case, we prove that there is an associated natural martingale Yn* converging to a non-negative random variable Z* with infinite mean. Both Z and Z* lead to non-trivial solution of a distributional equation which extends the notion of stable laws. Precise results are obtained about Hausdorff measures and packing measures of the support of the Mandelbrot's measure.

Suggested Citation

  • Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
    • Handle: RePEc:eee:spapps:v:86:y:2000:i:2:p:263-286
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