Journal
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
Volume 86, Issue 2, Pages 263-286Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/S0304-4149(99)00097-6
Keywords
self-similar cascades; marked trees; branching random walk; random measures; martingales; functional equations; moments; exponential moments; tails; Hausdorff measure; packing measure
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We consider a generalized Mandelbrot's martingale {Y-n} and the associated Mandelbrot's measure mu(alpha) on marked trees. If the limit variable Z = lim Y-n 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 Y-n* 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. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: Primary: 60J80, 60G57, 28A78, 28A80; Secondary: 60G42.
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