Limit theorems for multiplicative processes (Q1426860)

Let \(W\) be a random variable such that \(W\geq 0\), \(EW= 1\), and let \(\{W_{i_1\cdots i_n}\}\) be a family of independent copies of \(W\), indexed by all finite sequences \(i_1\cdots i_n\) of positive integers. For fixed \(r\) and \(n\) the random multiplicative measure \(\mu^n_r\) has, on each \(r\)-adic interval \(A^r_{i_1\cdots i_n}\) at \(n\)th level, the density \(W_{i_1}\cdots W_{i_1\cdots i_n}\) on \([0,1]\). For each fixed \(1\leq n\leq\infty\) the asymptotic properties (uniform law of large numbers, functional central limit theorem, large deviation principle) for the sequence of random measures \(\{\mu^n_r\}\) are studied as \(r\to\infty\).