On Normalized Multiplicative Cascades under Strong Disorder

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