Abstract
Let W be a non-negative random variable with EW=1, and let {W i } be a family of independent copies of W, indexed by all the finite sequences i=i 1⋅⋅⋅i n of positive integers. For fixed r and n the random multiplicative measure μ n r has, on each r-adic interval \(A_{i_1 ...i_n }^r \) at nth level, the density \(W_{i_1 } \cdot \cdot \cdot W_{i_1 \ldots i_n } \) with respect to the Lebesgue measure on [0,1]. If EW log W<log r, the sequence {μ n r } 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 {μ n r } 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 \(\{ \mu _r^n (f),\;f \in G\}\) 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 {μ 1 r } r , and the recent ones for the masses of {μ ∞ r } r established in Ref. 23.
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Liu, Q., Rio, E. & Rouault, A. Limit Theorems for Multiplicative Processes. Journal of Theoretical Probability 16, 971–1014 (2003). https://doi.org/10.1023/B:JOTP.0000012003.49768.f6
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DOI: https://doi.org/10.1023/B:JOTP.0000012003.49768.f6