Iksanov O. Fixed points of inhomogeneous smoothing transforms.

In the thesis stochastic fixed points of several inhomogeneous smoothing transforms are investigated. New techniques worked out in the thesis made it possible to study (а) properties of convergent perpetuities; (b) fixed points of homogeneous smoothing transforms for which the number of points in an underlying point process is not assumed to be almost surely finite; (в) fixed points of generalized shot noise transforms, without extra moment assumptions at all or under minimal such assumptions. In particular, a connection has been revealed between martingales related to branching random walks in which the number of children of one (each) individual is not assumed to be almost surely finite, and convergent perpetuities. This allowed us to solve several problems concerning these martingales and to develop a theory of fixed points of homogeneous smoothing transforms. We have investigated in detail a relation between self-decomposable distributions and shot noise distributions. Also we have discovered and characterized a new property of self-decomposable distributions which clarifies self-decomposability of some distributions arising in the stochastic analysis. Finally, a relation has been discovered between fixed points of generalized shot noise transforms and a Pitman-Yor problem which allowed us to solve the latter problem.