A VARIANT OF MANDELBROT,B. MARTINGALES

Let c be an integer greater than or equal to 2, W a r.v. such that E(W ) = 1/c, I-0 = [0, 1[ and F the set of c-adic strict subintervals of I -0, semi-open to the right. Let {W(I)}(I is an element of F) be a sequ ence of i.i.d. r.v.'s, with W(I) similar to W, and m(I) = Pi(I subset of J is an element of F) W(J). If (F-n)(n greater than or equal to 1) is a sequence of finer and finer finite partitions of I-0 by elements of F, we denote the sequence (Sigma(I is an element of Fn) m(I))(n gre ater than or equal to 1) by (Y-n)(n greater than or equal to 1). When F-n is the set of the I's of length c(-n), (Y-n)(n greater than or equ al to 1) is the Mandelbrot canonical martingale, studied [1], which co nverges a.s. We prove that in the general case where (F-n)(n greater t han or equal to 1) is random, independent of the W(I)'s, (Y-n)(n great er than or equal to 1) is a martingale which converges a.s. towards th e same limit as in. the canonical case.