On normalized multiplicative cascades under strong disorder

Partha Dey, Edward Waymire
2015 Electronic Communications in Probability  
Multiplicative cascades, under weak or strong disorder, refer to sequences of positive random measures µ n, , n = 1, 2, . . . , parameterized by a positive disorder parameter , and defined on the Borel -field B of @T = {0, 1, . . . b 1} 1 for the product topology. The normalized cascade is defined by the corresponding sequence of random probability measures prob n, := Z 1 n, µ n, , n = 1, 2 . . . , normalized to a probability by the partition function Z n, . In this note, a recent result of
more » ... ecent result of Madaule [27, 2011] is used to explicitly construct a family of tree indexed probability measures prob 1, for strong disorder parameters > c, almost surely defined on a common probability space. Moreover, viewing {prob n, : > c} 1 n=1 as a sequence of probability measure valued stochastic process leads to finite dimensional weak convergence in distribution to a probability measure valued process {prob 1, : > c}. The limit process is constructed from the tree-indexed random field of derivative martingales, and the Brunet-Derrida-Madaule decorated Poisson process. A number of corollaries are provided to illustrate the utility of this construction.
doi:10.1214/ecp.v20-3936 fatcat:ny2sa4bc6rgihofqsbnbuhvgzq