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On Normalized Multiplicative Cascades under Strong Disorder

Multiplicative cascades, under weak or strong disorder, refer to sequences of positive random measures $μ_{n,β}, n = 1,2,\dots$, parameterized by a positive disorder parameter $β$, and defined on the Borel $σ$-field ${\mathcal B}$ of $\partial T = \{0,1,\dots b-1\}^\infty$ for the product topology. The normalized cascade is defined by the corresponding sequence of random probability measures $prob_{n,β}:= Z_{n,β}^{-1}μ_{n,β}, n = 1,2\dots,$ normalized to a probability by the partition function $Z_{n,β}$. In this note, a recent result of Madaule (2011) is used to explicitly construct a family of tree indexed probability measures $prob_{\infty,β}$ for strong disorder parameters $β> β_c$, almost surely defined on a common probability space. Moreover, viewing $\{prob_{n,β}: β> β_c\}_{n=1}^\infty$ as a sequence of probability measure valued stochastic process leads to finite dimensional weak convergence in distribution to a probability measure valued process $\{prob_{\infty,β}: β> β_c\}$. 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.