arXiv subjects

Gerold Alsmeyer

Publications and source records attributed to Gerold Alsmeyer.

At least 37 records · Page 2Linked to original sources

arXiv2017

Stability of perpetuities in Markovian environment

The stability of iterations of affine linear maps $Ψ_{n}(x)=A_{n}x+B_{n}$, $n=1,2,\ldots$, is studied in the presence of a Markovian environment, more precisely, for the situation when $(A_{n},B_{n})_{n\ge 1}$ is modulated by an ergodic Markov chain $(M_{n})_{n\ge 0}$ with countable state space $\mathcal{S}$ and stationary distribution $π$. We provide necessary and sufficient conditions for the a.s. and the distributional convergence of the backward iterations $Ψ_{1}\circ\ldots\circΨ_{n}(Z_{0})$ and also describe all possible limit laws as solutions to a certain Markovian stochastic fixed-point equation. As a consequence of the random environment, these limit laws are stochastic kernels from $\mathcal{S}$ to $\mathbb{R}$ rather than distributions on $\mathbb{R}$, thus reflecting their dependence on where the driving chain is started. We give also necessary and sufficient conditions for the distributional convergence of the forward iterations $Ψ_{n}\circ\ldots\circΨ_{1}$. The main differences caused by the Markovian environment as opposed to the extensively studied case of independent and identically distributed (iid) $Ψ_{1},Ψ_{2},\ldots$ are that: (1) backward iterations may still converge in distribution, if a.s. convergence fails, (2) the degenerate case when $A_{1}c_{M_{1}}+B_{1}=c_{M_{0}}$ a.s. for suitable constants $c_{i}$, $i\in\mathcal{S}$, is by far more complex than the degenerate case for iid $(A_{n},B_{n})$ when $A_{1}c+B_{1}=c$ a.s. for some $c\in\mathbb{R}$, and (3) forward and backward iterations generally have different laws given $M_{0}=i$ for $i\in\mathcal{S}$ so that the former ones need a separate analysis. Our proofs draw on related results for the iid-case, notably by Vervaat, Grincevičius, and Goldie and Maller, in combination with recent results by the authors on fluctuation theory for Markov random walks.

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arXiv2016

A leader-election procedure using records

The study of the number of collisions in a Poisson-Dirichlet coalescent leads to the analysis of the following version of a stochastic leader-elec\-tion algorithm. Consider an infinite family of persons, labeled by $1,2,3,\ldots$, who generate iid random numbers from an arbitrary continuous distribution. Those persons who have generated a record value, that is, a value larger than the values of all previous persons, stay in the game, all others must leave. The remaining persons are relabeled by $1,2,3,\ldots$ maintaining their order in the first round, and the election procedure is repeated independently from the past and indefinitely. We prove limit theorems for a number of relevant functionals for this procedure, notably the number of rounds $T(M)$ until all persons among $1,\ldots,M$, except the first one, have left (as $M\to\infty$). For example, we show that the sequence $(T(M)-\log^{*}M)_{M\in\mathbb{N}}$, where $\log^{*}$ denotes the iterated logarithm, is tight, and study its weak subsequential limits. We further provide an appropriate and apparently new kind of normalization (based on tetrations) such that the original labels of persons who stay in the game until round $n$ converge (as $n\to\infty$) to some random non-Poissonian point process and study its properties. The results are applied to study subsequential distributional limits for the number of collisions in the Poisson-Dirichlet coalescent.

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arXiv2016

Leader election using random walks

In the classical leader election procedure all players toss coins independently and those who get tails leave the game, while those who get heads move to the next round where the procedure is repeated. We investigate a generalizion of this procedure in which the labels (positions) of the players who remain in the game are determined using an integer-valued random walk. We study the asymptotics of some relevant quantities for this model such as: the positions of the persons who remained after $n$ rounds; the total number of rounds until all the persons among $1,2,\ldots,M$ leave the game; and the number of players among $1,2,\ldots,M$ who survived the first $n$ rounds. Our results lead to some interesting connection with Galton-Watson branching processes and with the solutions of certain stochastic-fixed point equations arising in the context of the stability of point processes under thinning. We describe the set of solutions to these equations and thus provide a characterization of one-dimensional point processes that are stable with respect to thinning by integer-valued random walks.

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arXiv2016

Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

The Bernoulli sieve is the infinite Karlin "balls-in-boxes" scheme with random probabilities of stick-breaking type. Assuming that the number of placed balls equals $n$, we prove several functional limit theorems (FLTs) in the Skorohod space $D[0,1]$ endowed with the $J_{1}$- or $M_{1}$-topology for the number $K_{n}^{*}(t)$ of boxes containing at most $[n^{t}]$ balls, $t\in[0,1]$, and the random distribution function $K_{n}^{*}(t)/K_{n}^{*}(1)$, as $n\to\infty$. The limit processes for $K_{n}^{*}(t)$ are of the form $(X(1)-X((1-t)-))_{t\in[0,1]}$, where $X$ is either a Brownian motion, a spectrally negative stable Lévy process, or an inverse stable subordinator. The small values probabilities for the stick-breaking factor determine which of the alternatives occurs. If the logarithm of this factor is integrable, the limit process for $K_{n}^{*}(t)/K_{n}^{*}(1)$ is a Lévy bridge. Our approach relies upon two novel ingredients and particularly enables us to dispense with a Poissonization-de-Poissonization step which has been an essential component in all the previous studies of $K_{n}^{*}(1)$. First, for any Karlin occupancy scheme with deterministic probabilities $(p_{k})_{k\ge 1}$, we obtain an approximation, uniformly in $t\in[0,1]$, of the number of boxes with at most $[n^{t}]$ balls by a counting function defined in terms of $(p_{k})_{k\ge 1}$. Second, we prove several FLTs for the number of visits to the interval $[0,nt]$ by a perturbed random walk, as $n\to\infty$.

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arXiv2015

Ladder epochs and ladder chain of a Markov random walk with discrete driving chain

Let $(M_{n},S_{n})_{n\ge 0}$ be a Markov random walk with positive recurrent driving chain $(M_{n})_{n\ge 0}$ having countable state space $\mathcal{S}$ and stationary distribution $π$. It is shown in this note that, if the dual sequence $({}^{\#}M_{n},{}^{\#}S_{n})_{n\ge 0}$ is positive divergent, i.e. ${}^{\#}S_{n}\to\infty$ a.s., then the strictly ascending ladder epochs $σ_{n}^{>}$ of $(M_{n},S_{n})_{n\ge 0}$ are a.s. finite and the ladder chain $(M_{σ_{n}^{>}})_{n\ge 0}$ is positive recurrent on some $\mathcal{S}^{>}\subset\mathcal{S}$. We also provide simple expressions for its stationary distribution $π^{>}$, an extension of the result to the case when $(M_{n})_{n\ge 0}$ is null recurrent, and a counterexample that demonstrates that ${}^{\#}S_{n}\to\infty$ a.s. does not necessarily entail $S_{n}\to\infty$ a.s., but rather $\limsup_{n\to\infty}S_{n}=\infty$ a.s. only. Our arguments are based on Palm duality theory, coupling and the Wiener-Hopf factorization for Markov random walks with discrete driving chain.

math.PR↗

arXiv2015

Thin tails of fixed points of the nonhomogeneous smoothing transform

For a given random sequence $(C,T_{1},T_{2},\ldots)$ with nonzero $C$ and a.s. finite number of nonzero $T_{k}$, the nonhomogeneous smoothing transform $\mathcal{S}$ maps the law of a real random variable $X$ to the law of $\sum_{k\ge 1}T_{k}X_{k}+C$, where $X_{1},X_{2},\ldots$ are independent copies of $X$ and also independent of $(C,T_{1},T_{2},\ldots)$. This law is a fixed point of $\mathcal{S}$ if the stochastic fixed-point equation (SFPE) $X\stackrel{d}{=}\sum_{k\ge 1}T_{k}X_{k}+C$ holds true, where $\stackrel{d}{=}$ denotes equality in law. Under suitable conditions including $\mathbb{E} C=0$, $\mathcal{S}$ possesses a unique fixed point within the class of centered distributions, called the canonical solution to the above SFPE because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on $(C,T_{1},T_{2},\ldots)$ such that the canonical solution exhibits right and/or left Poisson tails and the abscissa of convergence of its moment generating function can be determined. As a particular application, the right tail behavior of the Quicksort distribution is found.

math.PR↗

arXiv2015

Renewal approximation for the absorption time of a decreasing Markov chain

We consider a Markov chain $(M_{n})_{n\ge 0}$ on the set $\mathbb{N}_{0}$ of nonnegative integers which is eventually decreasing, i.e. $\mathbb{P}\{M_{n+1}<M_{n}|M_{n}\ge a\}=1$ for some $a\in\mathbb{N}$ and all $n\ge 0$. We are interested in the asymptotic behaviour of the law of the stopping time $T=T(a):=\inf\{k\in\mathbb{N}_{0}: M_{k}<a\}$ under $\mathbb{P}_{n}:=\mathbb{P}(\cdot|M_{0}=n)$ as $n\to\infty$. Assuming that the decrements of $(M_{n})_{n\ge 0}$ given $M_{0}=n$ possess a kind of stationarity for large $n$, we derive sufficient conditions for the convergence in minimal $L^{p}$-distance of $\mathbb{P}_{n}((T-a_{n})/b_{n}\in\cdot)$ to some non-degenerate, proper law and give an explicit form of the constants $a_{n}$ and $b_{n}$.

math.PR↗

arXiv2015

On the stationary tail index of iterated random Lipschitz functions

Let $Ψ_1,Ψ_2,...$ be a sequence of i.i.d. random Lipschitz functions on a complete separable metric space with unbounded metric $d$ and forward iterations $X_n$. Suppose that $X_n$ has a stationary distribution. We study the stationary tail behavior of the functional $D_n=d(x_0,X_n)$, $x_0$ an arbitrary reference point, by providing bounds for these random variables in terms of simple contractive iterated function systems on the nonnegative halfline. Our results provide bounds for the lower and upper tail index of $D_n$ and will be illustrated by a number of popular examples including the AR(1) model with ARCH errors and random logistic transforms.

math.PR↗

arXiv2015

Branching within branching I: The extinction problem

We consider a discrete-time host-parasite model for a population of cells which are colonized by proliferating parasites. The cell population grows like an ordinary Galton-Watson process, but in reflection of real biological settings the multiplication mechanisms of cells and parasites are allowed to obey some dependence structure. More precisely, the number of offspring produced by a mother cell determines the reproduction law of a parasite living in this cell and also the way the parasite offspring is shared into the daughter cells. In this article, we provide a formal introduction of this branching-within-branching model and then focus on the property of parasite extinction. We establish equivalent conditions for almost sure extinction of parasites, and find a strong relation of this event to the behavior of parasite multiplication along a randomly chosen cell line through the cell tree, which forms a branching process in random environment. In a second paper, the case when parasites survive is studied by proving limit results.

math.PR↗

arXiv2015

Branching within branching II: Limit theorems

This continues work started in part I on a general branching-within-branching model for host-parasite co-evolution. Here we focus on asymptotic results for relevant processes in the case when parasites survive. In particular, limit theorems for the processes of contaminated cells and of parasites are established by using martingale theory and the technique of size-biasing. The results for both processes are of Kesten-Stigum type by including equivalent integrability conditions for the martingale limits to be positive with positive probability. The case when these conditions fail is also studied. For the process of contaminated cells, we show that a proper Heyde-Seneta norming exists such that the limit is nondegenerate.

math.PR↗

arXiv2014

Quasi-stochastic matrices and Markov renewal theory

Given a matrix of distribution functions and a quasi-stochastic matrix, i.e. an irreducible nonnegative matrix with maximal eigenvalue one and associated unique positive left and right eigenvectors, the article studies the properties of an associated matrix renewal measure and a related integral equation. Unlike earlier work this is done by a purely probabilistic approach based on a simple harmonic transform. Main results include Markov renewal-type theorems and a Stone-type decomposition under an absolute continuity condition. Three applications are given at the end of the paper.

math.PR↗

arXiv2013

A host-parasite model for a two-type cell population

A host-parasite model is considered for a population of cells that can be of two types, A or B, and exhibits unilateral reproduction: while a B-cell always splits into two cells of the same type, the two daughter cells of an A-cell can be of any type. The random mechanism that describes how parasites within a cell multiply and are then shared into the daughter cells is allowed to depend on the hosting mother cell as well as its daughter cells. Focusing on the subpopulation of A-cells and its parasites, the model differs from the single-type model recently studied by Bansaye (2008) in that the sharing mechanism may be biased towards one of the two types. Main results are concerned with the nonextinctive case and provide information on the behavior, as $n\to\infty$, of the number A-parasites in generation n and the relative proportion of A- and B-cells in this generation which host a given number of parasites. As in (Bansaye,2008), proofs will make use of a so-called random cell line which, when conditioned to be of type A, behaves like a branching process in random environment.

math.PR↗

arXiv2013

Power and exponential moments of the number of visits and related quantities for perturbed random walks

Let $(ξ_1,η_1),(ξ_2,η_2),...$ be a sequence of i.i.d.\ copies of a random vector $(ξ,η)$ taking values in $\R^2$, and let $S_n := ξ_1+...+ξ_n$. The sequence $(S_{n-1} + η_n)_{n \geq 1}$ is then called perturbed random walk. We study random quantities defined in terms of the perturbed random walk: $τ(x)$, the first time the perturbed random walk exits the interval $(-\infty,x]$, $N(x)$, the number of visits to the interval $(-\infty,x]$, and $ρ(x)$, the last time the perturbed random walk visits the interval $(-\infty,x]$. We provide criteria for the a.s.\ finiteness and for the finiteness of exponential moments of these quantities. Further, we provide criteria for the finiteness of power moments of $N(x)$ and $ρ(x)$.

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arXiv2012

Metabasins - a State Space Aggregation for highly disordered Energy Landscapes

Glass-forming systems, which are characterized by a highly disordered energy landscape, have been studied in physics by a simulation-based state space aggregation. The purpose of this article is to develop a path-independent approach within the framework of aperiodic, reversible Markov chains with exponentially small transition probabilities which depend on some energy function. This will lead to the definition of certain metastates, also called metabasins in physics. More precisely, our aggregation procedure will provide a sequence of state space partitions such that on an appropriate aggregation level certain properties (see Properties 1--4 of the Introduction) are fulfilled. Roughly speaking, this will be the case for the finest aggregation such that transitions back to an already visited (meta-)state are very unlikely within a moderate time frame.

math.PR↗

arXiv2012

The functional equation of the smoothing transform

Given a sequence $T=(T_i)_{i\geq1}$ of nonnegative random variables, a function f on the positive halfline can be transformed to $\mathbb{E}\prod_{i\geq1}f(tT_i)$. 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 $[0,\infty)$, the results provide the full description of the set of solutions to the fixed-point equation of the smoothing transform, $X\stackrel{d}{=}\sum_{i\geq1}T_iX_i$, where $\stackrel{d}{=}$ denotes equality of the corresponding laws, and $X_1,X_2,...$ is a sequence of i.i.d. copies of X independent of T. Further, since left-continuous survival functions are covered as well, the results also apply to the fixed-point equation $X\stackrel{d}{=}\inf\{X_i/T_i:i\geq1,T_i>0\}$. Moreover, we investigate the phenomenon of endogeny in the context of the smoothing transform and, thereby, solve an open problem posed by Aldous and Bandyopadhyay.

math.PR↗

arXiv2012

Precise tail index of fixed points of the two-sided smoothing transform

We consider real-valued random variables R satisfying the distributional equation R \eqdist \sum_{k=1}^{N}T_k R_k + Q, where R_1,R_2,... are iid copies of R and independent of T=(Q, (T_k)_{k \ge 1}). N is the number of nonzero weights T_k and assumed to be a.s. finite. Its properties are governed by the function m(s) := \E \sum_{k=1}^N |T_k|^s . There are at most two values α< βsuch that m(α)=m(β)=1. We consider solutions R with finite moment of order s > α. We review results about existence and uniqueness. Assuming the existence of βand an additional mild moment condition on the T_{k}, our main result asserts that \lim_{t \to \infty} t^βP(|R| > t) = K > 0, the main contribution being that K is indeed positive and therefore βthe precise tail index of |R|, for the convergence was recently shown by Jelenkovic and Olvera-Cravioto (arXiv:1012.2165).

math.PR↗

arXiv2011

Fixed points of the smoothing transform: Two-sided solutions

Given a sequence $(C,T) = (C,T_1,T_2,...)$ of real-valued random variables with $T_j \geq 0$ for all $j \geq 1$ and almost surely finite $N = \sup\{j \geq 1: T_j > 0\}$, the smoothing transform associated with $(C,T)$, defined on the set $\mathcal{P}(\R)$ of probability distributions on the real line, maps an element $P\in\mathcal{P}(\R)$ to the law of $C + \sum_{j \geq 1} T_j X_j$, where $X_1,X_2,...$ is a sequence of i.i.d.\ random variables independent of $(C,T)$ and with distribution $P$. We study the fixed points of the smoothing transform, that is, the solutions to the stochastic fixed-point equation $X_{1}\stackrel{\mathrm{d}}{=}C + \sum_{j \geq 1} T_j X_j$. By drawing on recent work by the authors with J.D.\;Biggins, a full description of the set of solutions is provided under weak assumptions on the sequence $(C,T)$. This solves problems posed by Fill and Janson \cite{FJ2000} and Aldous and Bandyopadhyay \cite{AB2005}. Our results include precise characterizations of the sets of solutions to large classes of stochastic fixed-point equations that appear in the asymptotic analysis of divide-and-conquer algorithms, for instance the \texttt{Quicksort} equation.

math.PR↗

arXiv2011

Fixed points of inhomogeneous smoothing transforms

We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation $X \stackrel{d}{=} C + \sum_{i \geq 1} T_i X_i$, where $\stackrel{d}{=}$ means equality in distribution, $(C,T_1,T_2,...)$ is a given sequence of non-negative random variables and $X_1,X_2,...$ is a sequence of i.i.d.\ copies of the non-negative random variable $X$ independent of $(C,T_1,T_2,...)$. 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. Further, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C=0. Using this correspondence, we present a full characterization of the set of fixed points under mild assumptions.

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