Poissonian products of random weights: Uniform convergence and related measures

  • Julien Barral

    Domaine de Voluceau, Le Chesnay, France


The random multiplicative measures on R\mathbb{R} introduced in Mandelbrot ([Mandelbrot 1996]) are a fundamental particular case of a larger class we deal with in this paper. An element μ\mu of this class is the vague limit of a continuous time measure-valued martingale μt\mu _{t}, generated by multiplying i.i.d. non-negative random weights, the (WM)MS(W_M)_{M\in S}, attached to the points MM of a Poisson point process SS, in the strip H={(x,y)R×R+;0<y1}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 (μλ)λU(\mu ^{\lambda})_{\lambda \in U} of such measures constructed simultaneously, when every measure μλ\mu^{\lambda} is obtained from a family of random weights (WM(λ))MS(W_M(\lambda))_{M\in S} and WM(λ)W_M(\lambda) depends smoothly upon the parameter λUR\lambda\in U\subset\mathbb{R}. This problem leads to study in several sense the convergence, for every s0s\geq 0, of the functions valued martingale Zt(s):λμtλ([0,s])Z^{(s)}_t: \lambda \mapsto \mu_{t}^{\lambda }([0,s]). The study includes the case of analytic versions of Zt(s)(λ)Z^{(s)}_t(\lambda) where λCn\lambda\in\mathbb{C}^n. The results make it possible to show in certain cases that the dimension of μλ\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 sμ([0,s])s\mapsto \mu ([0,s]) for which we derive limit theorems, with uniform versions when μ\mu depends on λ\lambda.

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