The random multiplicative measures on introduced in Mandelbrot ([Mandelbrot 1996]) are a fundamental particular case of a larger class we deal with in this paper. An element of this class is the vague limit of a continuous time measure-valued martingale , generated by multiplying i.i.d. non-negative random weights, the , attached to the points of a Poisson point process , in the strip 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 of such measures constructed simultaneously, when every measure is obtained from a family of random weights and depends smoothly upon the parameter . This problem leads to study in several sense the convergence, for every , of the functions valued martingale . The study includes the case of analytic versions of where . The results make it possible to show in certain cases that the dimension of 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 for which we derive limit theorems, with uniform versions when depends on .
Cite this article
Julien Barral, Poissonian products of random weights: Uniform convergence and related measures. Rev. Mat. Iberoam. 19 (2003), no. 3, pp. 813–856DOI 10.4171/RMI/371