Papers by Gerold Alsmeyer
Teubner-Skripten zur mathematischen Stochastik, 1991
arXiv (Cornell University), Oct 21, 2015
For a given random sequence (C, T 1 , T 2 ,. . .) with nonzero C and a.s. finite number of nonzer... more For a given random sequence (C, T 1 , T 2 ,. . .) with nonzero C and a.s. finite number of nonzero T k , the nonhomogeneous smoothing transform S maps the law of a real random variable X to the law of k≥1 T k X k + C, where X 1 , X 2 ,. .. are independent copies of X and also independent of (C, T 1 , T 2 ,. . .). This law is a fixed point of S if the stochastic fixed-point equation (SFPE) X d = k≥1 T k X k + C holds true, where d = denotes equality in law. Under suitable conditions including EC = 0 (see (10)), 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 ,. . .) 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.
arXiv (Cornell University), Nov 17, 2015
Let (M n , S n) n≥0 be a Markov random walk with positive recurrent driving chain (M n) n≥0 havin... more Let (M n , S n) n≥0 be a Markov random walk with positive recurrent driving chain (M n) n≥0 having countable state space S and stationary distribution π. It is shown in this note that, if the dual sequence (# M n , # S n) n≥0 is positive divergent, i.e. # S n → ∞ a.s., then the strictly ascending ladder epochs σ > n of (M n , S n) n≥0 (see (3)) are a.s. finite and the ladder chain (M σ > n) n≥0 is positive recurrent on some S > ⊂ S. We also provide simple expressions for its stationary distribution π > , an extension of the result to the case when (M n) n≥0 is null recurrent, and a counterexample that demonstrates that # S n → ∞ a.s. does not necessarily entail S n → ∞ a.s., but rather lim sup n→∞ S n = ∞ 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.
arXiv (Cornell University), Sep 13, 2010
Given a sequence (C, T) = (C, T 1 , T 2 ,. . .) of real-valued random variables with T j ≥ 0 for ... more Given a sequence (C, T) = (C, T 1 , T 2 ,. . .) of real-valued random variables with T j ≥ 0 for all j ≥ 1 and almost surely finite N = sup{j ≥ 1 : T j > 0}, the smoothing transform associated with (C, T), defined on the set P(R) of probability distributions on the real line, maps an element P ∈ P(R) to the law of C + j≥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 d = C + j≥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 [15] and Aldous and Bandyopadhyay [1]. 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 Quicksort equation.
arXiv (Cornell University), Jan 26, 2018
Let Ln be the least common multiple of a random set of integers obtained from {1,. .. , n} by ret... more Let Ln be the least common multiple of a random set of integers obtained from {1,. .. , n} by retaining each element with probability θ ∈ (0, 1) independently of the others. We prove that the process (log L nt) t∈[0,1] , after centering and normalization, converges weakly to a certain Gaussian process that is not Brownian motion. Further results include a strong law of large numbers for log Ln as well as Poisson limit theorems in regimes when θ depends on n in an appropriate way.
arXiv (Cornell University), Apr 2, 2012
We consider a host-parasite model for a population of cells that can be of two types, A or B, and... more We consider a host-parasite model 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, our model differs from the single-type model recently studied by Bansaye [5] in that the sharing mechanism may be biased towards one of the two types. Our main results are concerned with the nonextinctive case and provide information on the behavior, as n → ∞, 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 [5], 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.
Stochastic Processes and their Applications, Mar 1, 2002
This article continues work in [4] on random walks (S n) n≥0 whose increments X n are (m + 1)-blo... more This article continues work in [4] on random walks (S n) n≥0 whose increments X n are (m + 1)-block factors of the form ϕ(Y n−m , ..., Y n) for i.i.d. random variables Y −m , Y −m+1 , ... taking values in an arbitrary measurable space (S, S). Defining M n = (Y n−m , ..., Y n) for n ≥ 0, which is a Harris ergodic Markov chain, the sequence (M n , S n) n≥0 constitutes a Markov random walk with stationary drift µ = E F m+1 X 1 where F denotes the distribution of the Y n 's. Suppose µ > 0, let (σ n) n≥0 be the sequence of strictly ascending ladder epochs associated with (M n , S n) n≥0 and let (M σ n , S σ n) n≥0 , (M σ n , σ n) n≥0 be the resulting Markov renewal processes whose common driving chain is again positive Harris recurrent. The Markov renewal measures associated with (M n , S n) n≥0 and the former two sequences are denoted U λ , U > λ and V > λ , respectively, where λ is an arbitrary initial distribution for (M 0 , S 0). Given the basic sequence (M n , S n) n≥0 is spread-out or 1-arithmetic with shift function 0, we provide convergence rate results for each of U λ , U > λ and V > λ under natural moment conditions. Proofs are based on a suitable reduction to standard renewal theory by finding an appropriate imbedded regeneration scheme and coupling. Considerable work is further spent on necessary moment results.
Stochastic Processes and their Applications, Jun 1, 2016
We consider a discrete-time host-parasite model for a population of cells which are colonized by ... more 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. We then 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. c
arXiv (Cornell University), Aug 30, 2016
Two fundamental theorems by Spitzer-Erickson and Kesten-Maller on the fluctuation type (positive ... more Two fundamental theorems by Spitzer-Erickson and Kesten-Maller on the fluctuation type (positive divergence, negative divergence or oscillation) of a real-valued random walk (S n) n≥0 with iid increments X 1 , X 2 ,. .. and the existence of moments of various related quantities like the first passage into (x, ∞) and the last exit time from (−∞, x] for arbitrary x ≥ 0 are studied in the Markov-modulated situation when the X n are governed by a positive recurrent Markov chain M = (M n) n≥0 on a countable state space S, thus for a Markov random walk (M n , S n) n≥0. Our approach is based on the natural strategy to draw on the results in the iid case for the embedded ordinary random walks (S τn(i)) n≥0 , where τ 1 (i), τ 2 (i),. .. denote the successive return times of M to state i, and an analysis of the excursions of the walk between these epochs. However, due to these excursions, generalizations of the afore-mentioned theorems are surprisingly more complicated and require the introduction of various excursion measures so as to characterize the existence of moments of different quantities.
Journal of Theoretical Probability, Feb 24, 2023
The concept of homology, originally developed as a useful tool in algebraic topology, has by now ... more The concept of homology, originally developed as a useful tool in algebraic topology, has by now become pervasive in quite different branches of mathematics. The notion particularly appears quite naturally in ergodic theory in the study of measurepreserving transformations arising from various group actions or, equivalently, the study of stationary sequences when adopting a probabilistic perspective as in this paper. Our purpose is to give a new and relatively short proof of the coboundary theorem due to Schmidt (Cocycles on ergodic transformation groups. Macmillan lectures in mathematics, vol 1, Macmillan Company of India, Ltd., Delhi, 1977) which provides a sharp criterion that determines (and rules out) when two stationary processes belong to the same null-homology equivalence class. We also discuss various aspects of nullhomology within the class of Markov random walks and compare null-homology with a formally stronger notion which we call strict-sense null-homology. Finally, we also discuss some concrete cases where the notion of null-homology turns up in a relevant manner.
Теория вероятностей и ее применения, 2005
arXiv (Cornell University), Dec 11, 2013
Given a matrix of distribution functions and a quasi-stochastic matrix, i.e. an irreducible nonne... more 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.
Theory of Probability and Its Applications, 2005
arXiv (Cornell University), Sep 30, 2021
Linear fractional Galton-Watson branching processes in i.i.d. random environment are, on the quen... more Linear fractional Galton-Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s. convergent counterpart in the positive recurrent case given by the corresponding backward iterations. The present expository article aims to provide an explicit view at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented here are known, we hope that the offered perspective will be welcomed by some readers.
Discrete Mathematics & Theoretical Computer Science, 2008
Gantert and Müller (2006) proved that a critical branching random walk (BRW) on the integer latti... more Gantert and Müller (2006) proved that a critical branching random walk (BRW) on the integer lattice is transient by analyzing this problem within the more general framework of branching Markov chains and making use of Lyapunov functions. The main purpose of this note is to show how the same result can be derived quite elegantly and even extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of certain associated random weighted location measures which, upon taking expectations, provide a useful connection to the well established theory of ordinary random walks with i.i.d. increments. A brief discussion of the asymptotic behavior of the left-and rightmost particles in a critical BRW as time goes to infinity is provided in the final section by drawing on recent work by Hu and Shi (2008).
Journal of Theoretical Probability, Apr 9, 2008
Statistics & Probability Letters, Nov 1, 2001
Let (S n) n≥0 be a zero-delayed nonarithmetic random walk with positive drift µ and (ξ n) n≥0 be ... more Let (S n) n≥0 be a zero-delayed nonarithmetic random walk with positive drift µ and (ξ n) n≥0 be a slowly varying perturbation process (see conditions (C.1-3) in the Introduction). The results of this note are two weak convergence theorems for the difference τ (t) − ν(t), as t → ∞, where τ (t) = inf{n ≥ 1 : S n > t} and ν(t) = inf{n ≥ 1 : S n + ξ n > t} denotes its nonlinear counterpart. The main result (Theorem 1) states the existence of a limit distribution for τ (t) − ν(t) providing the weak convergence of the ξ n to a distribution Λ. Two applications in sequential statistics are also given.
Springer eBooks, 2013
E. Le Page: Tails of a stationary probability measure for an affine stochastic recursion on the l... more E. Le Page: Tails of a stationary probability measure for an affine stochastic recursion on the line.- Yv. Guivarc'h: On homogeneity at infinity of stationary measures for affine stochastic recursions.- M. Stolz: Limit theorems for random elements of the compact classical groups.- T. Kriecherbauer: Universality of local eigenvalue statistics.- R. Speicher: Asymptotic eigenvalue distribution of random matrices and free stochastic analysis.- M. Peigne: Conditioned random walk in Weyl chambers and renewal theory in a cone.- D. Buraczewski: The linear stochastic equation R =_d \sum_{ i=1}^N A_iR_i + B in the critical case.- J. Collamore: Tail estimates for stochastic fixed point equations.- S. Mentemeier: On multivariate random difference equations.- M. Olvera-Cravioto: Tail asymptotics for solutions of stochastic fixed point equations on trees.- E. Damek: On fixed points of generalized multidimensional affine recursions.- G. Alsmeyer: The functional equation of the smoothing transform.- O. Friesen, M. Loewe: Limit theorems for the eigenvalues of random matrices with weakly correlated entries.
arXiv (Cornell University), Jul 2, 2009
This paper is devoted to the study of the stochastic fixed-point equation X d = inf i≥1:Ti>0 X i ... more This paper is devoted to the study of the stochastic fixed-point equation X d = inf i≥1:Ti>0 X i /T i and the connection with its additive counterpart X d = i≥1 T i X i associated with the smoothing transformation. Here d = means equality in distribution, T def = (T i) i≥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 [16].
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Papers by Gerold Alsmeyer