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 $H=\{(x,y)\in \mathbb{R}\times {\mathbb{R}}_{+};0<y\le 1\}$ 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 $s\ge 0$, of the functions valued martingale $Z_t^{(s)}:\lambda \mapsto {\mu }_t^{\lambda }([0,s])$. The study includes the case of analytic versions of $Z_t^{(s)}(\lambda )$ 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 $\lambda$.