November 27, 2013 at 14.10
There is considerable interest and a vast literature on generalizations of the concept of stability of distributions. One extension of this concept is that of random variables ``stable by random weighted mean'' (this notion is due to Liu). A random variable \(X\) taking values in \(\mathbb{R}^d\) is called "stable by random weighted mean'' if it satisfies a recursive distributional equation of the following type:
\begin{equation}X\stackrel{\mathcal{D}}{=}C+\sum_{j\geq 1}T_jX_j\hskip{2cm}(1)\end{equation}
Here, "\(\stackrel{\mathcal{D}}{=}\)'' denotes equality of the corresponding distributions, \((C,T_1,T_2,\ldots)\) is a given sequence of real-valued random variables, and \(X_1, X_2, \ldots\) denotes a sequence of i.i.d. copies of the random variable \(X\) that are independent of \((C,T_1,T_2,\ldots)\). The distributions \(P\) on \(\mathbb{R}^d\) such that (1) holds when \(X\) has distribution \(P\) are called fixed points of the smoothing transform (associated with \((C,T_1,T_2,\ldots)\)).
In my talk, I will give an overview over the study of (1) comprising the origins of the study, applications and recent developments.
Keywords: Branching random walks, distributional fixed-point equation; infinite divisibility; multiplicative martingales; smoothing transformation; stable distribution; weighted branching process.
Speaker: Meiners Matthias (Technical University of Darmstadt, Germany)
Date: November 27, 2013 at 14.10, auditorium 221.
