The Fixed Points of the Multivariate Smoothing Transform

The class of multivariate stable laws on the positive cone [0, ∞) d is very rich: the Levy measure can be decomposed in a radial and spherical measure, the latter of which may be any finite measure on the intersection of the unit sphere with the positive cone. What additional information is needed in order to identify the spherical measure uniquely?

Consider a multitype branching process in random environment, where at each branching instant, the children’s type is drawn according to a transition matrix with random entries, which depend on the environment. How does the average distribution of types evolve in time?

In order to address (and answer!) these questions, we study the folllowing stochastic fixed point equation:
X =(in law) T_1 X_1 + . . . + T_N X_N ,                                       (1)
where (T_1 , . . . , T_N ) are (given) random matrices with nonnegative entries, and X, X_1 , . . . , X_N are independent identically distributed random vectors. The aim is to determine what are the possible distributions for X.

In my talk, I will first discuss the two examples mentioned above and then describe in detail, how one can find solutions to (1). Products of random matrices will play an important role.