Poissonian products of random weights: Uniform convergence and related measures

Abstract

The random multiplicative measures on $\mathbb{R}$ introduced in Mandelbrot ([Mandelbrot 1996]) are a fundamental particular case of a larger class we deal with in this paper. An element $\mu$ of this class is the vague limit of a continuous time measure-valued martingale $\mu _{t}$, generated by multiplying i.i.d. non-negative random weights, the $(W_M)_{M\in S}$, attached to the points $M$ of a Poisson point process $S$, in the strip $H=\{(x,y)\in \mathbb{R}\times\mathbb{R}_+ ; 0 < y\leq 1\}$ of the upper half-plane. We are interested in giving estimates for the dimension of such a measure. Our results give these estimates almost surely for uncountable families $(\mu ^{\lambda})_{\lambda \in U}$ of such measures constructed simultaneously, when every measure $\mu^{\lambda}$ is obtained from a family of random weights $(W_M(\lambda))_{M\in S}$ and $W_M(\lambda)$ depends smoothly upon the parameter $\lambda\in U\subset\mathbb{R}$. This problem leads to study in several sense the convergence, for every $s\geq 0$, of the functions valued martingale $Z^{(s)}_t: \lambda \mapsto \mu_{t}^{\lambda }([0,s])$. The study includes the case of analytic versions of $Z^{(s)}_t(\lambda)$ where $\lambda\in\mathbb{C}^n$. The results make it possible to show in certain cases that the dimension of $\mu^{\lambda}$ depends smoothly upon the parameter. When the Poisson point process is statistically invariant by horizontal translations, this construction provides the new non-decreasing multifractal processes with stationary increments $s\mapsto \mu ([0,s])$ for which we derive limit theorems, with uniform versions when $\mu$ depends on $\lambda$.