“On certain martingales of Benoit Mandelbrot” Guest contribution (Kahane & Peyrière 1976)

Following his critical analysis of the random model of turbulence due to A, M. Yaglom, M 1974f{N14} and M1974c{N15} introduced his own model, which he calls “canonical.” It proceeds from a brick, that is subsequently divided into b, b 2, …, b n , … similar bricks; each brick of the n-th stage is divided into b equal bricks in the (n + l)-th stage. Also given is a sequence of random variables W p , which are independent, identically distributed, positive, have mean 1 and are indexed by the bricks P under consideration. Starting from the Lebesgue measure μ0 on the initial brick, one constructs the sequence of measures μ n by successive stages. Thus, μ n has a constant density on each brick P of the n-th stage, and the density of μ n on P is the product of W p and the density μ n−1 on P. The sequence of measures μ n is a vector martingale, and it converges towards a random measure μ. M 1974c gives results and raises problems concerning the measure μ: non-degeneracy, the moments of ∥μ∥, the Borel sets supporting μ and their Hausdorff dimension. Some of the conjectures of Mandelbrot have been solved by Kahane 1974 or by Peyrière 1974. Here we present these results in a refined form. Theorems 1, 2 and 3 below are due to J.-P. Kahane, Theorem 4 is due to J. Peyrière.