We show that for every positive p, the Lp-norm of linear combinations (with scalar or vector coefficients) of products of i.i.d. random variables, whose moduli have a nondegenerate distribution with the p-norm one, is comparable to the lp-norm of the coefficients and the constants are explicit....

Let N1 be a fixed integer and (C1,…,CN,Q) a random element of M(d×d,R)N×Rd. We consider solutions of multivariate smoothing transforms, i.e. random variables R satisfying R=d∑i=1NCiRi+Q where =d denotes equality in distribution, and R,R1,…,RN are independent identically distributed...