Matthias Meiners

The functional equation of the smoothing transform

Given a sequence T = (Ti)i� 1 of non-negative random variables, a function f on the positive halfline can be transformed to E Q i� 1 f(tTi). We study the fixed points of this transform within the class of decreasing functions. By exploiting the intimate relationship with general branching processes, a full description of the set of solutions is established without the moment conditions that figure in earlier studies. Since the class of functions under consideration contains all Laplace transforms of probability distributions on [0,1 ), the results provide the full description of the set of solutions to the fixed-point equation of the smoothing transform, X d

Fixed Points of Inhomogeneous Smoothing Transforms

We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation , where means equality in distribution, is a given sequence of non-negative random variables and is a sequence of i.i.d. copies of the non-negative random variable X independent of . In this situation, X (or, more precisely, the distribution of X) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necessary and sufficient condition for the existence of a fixed point. Furthermore, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C = 0. Using this correspondence and the known theory on the homogeneous equation, we present a full characterization of the set of fixed points under mild assumptions.

Asymptotics of random processes with immigration II: Convergence to stationarity

Let

Rate of convergence in the law of large numbers for supercritical general multi-type branching processes

X_1, X_2,\ldots

Convergence of complex martingales in the branching random walk: the boundary

be random elements of the Skorokhod space

Regularity of the Speed of Biased Random Walk in a One-Dimensional Percolation Model

D(\mathbb{R})

Fixed points of the smoothing transform: two-sided solutions

and

Power and Exponential Moments of the Number of Visits and Related Quantities for Perturbed Random Walks

\xi_1, \xi_2, \ldots

Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

positive random variables such that the pairs

Limit theorems for renewal shot noise processes with eventually decreasing response functions

(X_1,\xi_1), (X_2,\xi_2),\ldots