Limit Theorems for Multiplicative Processes

Let W be a non-negative random variable with EW=1, and let {Wi} be a family of independent copies of W, indexed by all the finite sequences i=i1⋅⋅⋅in of positive integers. For fixed r and n the random multiplicative measure μnr has, on each r-adic interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$A_{i_1 ...i_n }^r $$ \end{document} at nth level, the density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$W_{i_1 } \cdot \cdot \cdot W_{i_1 \ldots i_n } $$ \end{document} with respect to the Lebesgue measure on [0,1]. If EW log W<log r, the sequence {μnr}n converges a.s. weakly to the Mandelbrot measure μ∞r. For each fixed 1≤n≤∞, we study asymptotic properties for the sequence of random measures {μnr}r as r→∞. We prove uniform laws of large numbers, functional central limit theorems, a functional law of iterated logarithm, and large deviation principles. The function-indexed processes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\{ \mu _r^n (f),\;f \in G\}$$ \end{document} is a natural extension to a tree-indexed process at nth level of the usual smoothed partial-sum process corresponding to n=1. The results extend the classical ones for {μ1r}r, and the recent ones for the masses of {μ∞r}r established in Ref. 23.