# Poissonian products of random weights: Uniform convergence and related measures

### Julien Barral

Domaine de Voluceau, Le Chesnay, France

## Abstract

The random multiplicative measures on $R$ introduced in Mandelbrot ([Mandelbrot 1996]) are a fundamental particular case of a larger class we deal with in this paper. An element $μ$ of this class is the vague limit of a continuous time measure-valued martingale $μ_{t}$, generated by multiplying i.i.d. non-negative random weights, the $(W_{M})_{M∈S}$, attached to the points $M$ of a Poisson point process $S$, in the strip $H={(x,y)∈R×R_{+};0<y≤1}$ 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 $(μ_{λ})_{λ∈U}$ of such measures constructed simultaneously, when every measure $μ_{λ}$ is obtained from a family of random weights $(W_{M}(λ))_{M∈S}$ and $W_{M}(λ)$ depends smoothly upon the parameter $λ∈U⊂R$. This problem leads to study in several sense the convergence, for every $s≥0$, of the functions valued martingale $Z_{t}:λ↦μ_{t}([0,s])$. The study includes the case of analytic versions of $Z_{t}(λ)$ where $λ∈C_{n}$. The results make it possible to show in certain cases that the dimension of $μ_{λ}$ 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 $s↦μ([0,s])$ for which we derive limit theorems, with uniform versions when $μ$ depends on $λ$.

## Cite this article

Julien Barral, Poissonian products of random weights: Uniform convergence and related measures. Rev. Mat. Iberoam. 19 (2003), no. 3, pp. 813–856

DOI 10.4171/RMI/371