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Gerold Alsmeyer

Publications and source records attributed to Gerold Alsmeyer.

42 records · Page 3Linked to original sources
arXiv2011

Tail behavior of stationary solutions of random difference equations: the case of regular matrices

Given a sequence $(M_{n},Q_{n})_{n\ge 1}$ of i.i.d. random variables with generic copy $(M,Q)$ such that $M$ is a regular $d\times d$ matrix and $Q$ takes values in $\mathbb{R}^{d}$, we consider the random difference equation (RDE) $R_{n}=M_{n}R_{n-1}+Q_{n}$, $n\ge 1$. Under suitable assumptions, this equation has a unique stationary solution $R$ such that, for some $κ>0$ and some finite positive and continuous function $K$ on $S^{d-1}:=\{x \in \mathbb{R}^{d}:|x|=1\}$, $ \lim_{t \to \infty} t^κ P(xR>t)=K(x)$ for all $x \in S^{d-1} $ holds true. This result is originally due to Kesten and Le Page. The purpose of this article is to show how regeneration methods can be used to provide a much shorter argument (in particular for the positivity of K). It is based on a multidimensional extension of Goldie's implicit renewal theory.

math.PR↗
arXiv2009

A min-type stochastic fixed-point equation related to the smoothing transformation

This paper is devoted to the study of the stochastic fixed-point equation X \stackrel{d}{=} \inf_{i \geq 1: T_i > 0} X_i/T_i and the connection with its additive counterpart $X \stackrel{d}{=} \sum_{i\ge 1}T_{i}X_{i}$ associated with the smoothing transformation. Here $\stackrel{d}{=}$ means equality in distribution, $T := (T_i)_{i \geq 1}$ is a given sequence of nonnegative random variables and $X, X_1, ...$ is a sequence of nonnegative i.i.d. random variables independent of $T$. We draw attention to the question of the existence of nontrivial solutions and, in particular, of special solutions named $α$-regular solutions $(α>0)$. We give a complete answer to the question of when $α$-regular solutions exist and prove that they are always mixtures of Weibull distributions or certain periodic variants. We also give a complete characterization of all fixed points of this kind. A disintegration method which leads to the study of certain multiplicative martingales and a pathwise renewal equation after a suitable transform are the key tools for our analysis. Finally, we provide corresponding results for the fixed points of the related additive equation mentioned above. To some extent, these results have been obtained earlier by Iksanov.

math.PR↗
arXiv2008

A stochastic fixed point equation for weighted minima and maxima

Given any finite or countable collection of real numbers $T_j,j\in J$, we find all solutions $F$ to the stochastic fixed point equation \[W\stackrel{\mathrm {d}}{=}\inf_{j\in J}T_jW_j,\] where $W$ and the $W_j,j\in J$, are independent real-valued random variables with distribution $F$ and $\stackrel{\mathrm {d}}{=}$ means equality in distribution. The bulk of the necessary analysis is spent on the case when $|J|\geq 2$ and all $T_j$ are (strictly) positive. Nontrivial solutions are then concentrated on either the positive or negative half line. In the most interesting (and difficult) situation $T$ has a characteristic exponent $α$ given by $\sum_{j\in J}T_j^α=1$ and the set of solutions depends on the closed multiplicative subgroup of $\mathbb {R}^{>}=(0,\infty)$ generated by the $T_j$ which is either $\{1\}$, $\mathbb {R}^{>}$ itself or $r^{\mathbb {Z}}=\{r^n\dvt n\in \mathbb {Z}\}$ for some $r>1$. The first case being trivial, the nontrivial fixed points in the second case are either Weibull distributions or their reciprocal reflections to the negative half line (when represented by random variables), while in the third case further periodic solutions arise. Our analysis builds on the observation that the logarithmic survival function of any fixed point is harmonic with respect to $\varLambda =\sum_{j\geq 1}δ_{T_j}$, i.e. $\varGamma =\varGamma \star \varLambda$, where $\star$ means multiplicative convolution. This will enable us to apply the powerful Choquet--Deny theorem.

math.PR↗
arXiv2008

A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks

Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection, first observed by the second author in [Iksanov, A.M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Proc. Appl. 114, 27-50.], arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons in [Lyons, R.(1997). A simple path to Biggins' martingale convergence for branching random walk. In Athreya, K.B., Jagers, P. (eds.). Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications, vol. 84, Springer, Berlin, 217-221.]. We also provide a necessary and sufficient condition for uniform integrability of these martingales in the most general situation which particularly means that the classical (LlogL)-condition is not always needed.

math.PR↗
arXiv2008

On distributional properties of perpetuities

We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of $p$-moments, $p>0$ as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.

math.PR↗