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A famous “curious identity” of Wallis gives a representation of the constant π in terms of a simply structured infinite product of fractions. Sondow and Yi [Amer. Math. Monthly 117(2010) 912–917] identified a general scheme for evaluating Wallis-type infinite products. The main purpose of this paper is to discuss an interpretation of the scheme by means of Pólya urn models.

A discrete-time Markov chain can be transformed into a new Markov chain by looking at its states along iterations of an almost surely finite stopping time. By the optional stopping theorem, any bounded harmonic function with respect to the transition function of the original chain is harmonic with respect to the transition function of the transformed chain. The reverse inclusion is in general not true. Our main result provides a sufficient condition on the stopping time which guarantees that the space of bounded harmonic functions for the transformed chain embeds in the space of bounded harmonic sequences for the original chain. We also obtain a similar result on positive unbounded harmonic functions under some additional conditions. Our work was motivated by and is analogous to the well-studied case when the Markov chain is a random walk on a discrete group.

In this paper, we formulate and analyze a Markov process modeling the motion of DNA nanomechanical walking devices. We consider a molecular biped restricted to a well-defined one-dimensional track and study its asymptotic behavior. Our analysis allows for the biped legs to be of different molecular composition, and thus to contribute differently to the dynamics. Our main result is a functional central limit theorem for the biped with an explicit formula for the effective diffusivity coefficient in terms of the parameters of the model. A law of large numbers, a recurrence/transience characterization and large deviations estimates are also obtained. Our approach is applicable to a variety of other biological motors such as myosin and motor proteins on polymer filaments.

A Quasi-Stationary Distribution (QSD)for a Markov process with an almost surely hit absorbing state is a time-invariant initial distribution for the process conditioned on not being absorbed by any given time. An initial distribution for the process is in the domain of attraction of some QSD \\(\\nu\\) if the distribution of the process a time \\(t\\), conditioned not to be absorbed by time \\(t\\) converges to \\(\\nu\\). In this work study mostly Brownian motion with constant drift on the half line \\([0,\\infty)\\) absorbed at \\(0\\). Previous work by Martinez et al. identifies all QSDs and provides a nearly complete characterization for their domain of attraction. Specifically, it was shown that if the distribution a well-defined exponential tail (including the case of lighter than any exponential tail), then it is in the domain of attraction of a QSD determined by the exponent. In this work we 1. Obtain a new approach to existing results, explaining the direct relation between a QSD and an initial distribution in its domain of attraction. 2. Study the behavior under a wide class of initial distributions whose tail is heavier than exponential, and obtain no-trivial limits under appropriate scaling.

This work provides complete description of Quasistationary Distributions (QSDs) for Markov chains with a unique absorbing state and an irreducible set of non-absorbing states. As is well-known, every QSD has an associated absorption parameter describing the exponential tail of the absorption time under the law of the process with the QSD as the initial distribution. The analysis associated with the existence and representation of QSDs corresponding to a given parameter is according to whether the moment generating function of the absorption time starting from any non-absorbing state evaluated at the parameter is finite or infinite, the finite or infinite moment generating function regimes, respectively. For parameters in the finite regime, it is shown that when exist, all QSDs are in the convex cone of a Martin entry boundary associated with the parameter. The infinite regime corresponds to at most one parameter value and at most one QSD. In this regime, when a QSD exists, it is unique and can be represented by a renewal-type formula. Multiple applications to the findings are presented, including revisiting some of the main classical results in the area.

We consider a model for which every site of \\(\\mathbb{N}\\) is assigned a fitness in \\([0,1]\\). At every discrete time all the sites are updated and each site samples a uniform on \\([0,1]\\), independently of everything else. At every discrete time and independently of the past the environment is good with probability \\(p\\) or bad with probability \\(1-p\\). The fitness of each site is then updated to the maximum or the minimum between its present fitness and the sampled uniform, according to whether the environment is good or bad. Assuming the initial fitness distribution is exchangeable over the site indexing, the empirical fitness distribution is a probability-valued Markov process. We show that this Markov process converges to an explicitly-identified stationary distribution exhibiting a self-similar structure.

A discrete-time Markov chain can be transformed into a new Markov chain by looking at its states along iterations of an almost surely finite stopping time. By the optional stopping theorem, any bounded harmonic function with respect to the transition function of the original chain is harmonic with respect to the transition function of the transformed chain. The reverse inclusion is in general not true. Our main result provides a sufficient condition on the stopping time which guarantees that the space of bounded harmonic functions for the transformed chain embeds in the space of bounded harmonic sequences for the original chain. We also obtain a similar result on positive unbounded harmonic functions, under some additional conditions. Our work was motivated by and is analogous to Forghani-Kaimanovich, the well-studied case when the Markov chain is a random walk on a discrete group.

We introduce a population model to test the hypothesis that even a single migrant per generation may rescue a dying population. Let \\((c_k)\\) be a sequence of real numbers in \\((0,1)\\). Let \\(X_n\\) be a size of the population at time \\(n\\geq 0\\). Then, \\(X_{n+1}=X_n - Y_{n+1}+1\\), where the conditional distribution of \\(Y_{n+1}\\) given \\(X_n=k\\) is a binomial random variable with parameters \\((k ,c(k))\\). We assume that \\(\\lim_{k\\to\\infty}kc(k)=\\rho\\) exists. If \\(\\rho<1\\) the process is transient with speed \\(1-\\rho\\) (so yes a single migrant per generation may rescue a dying population!) and if \\(\\rho>1\\) the process is positive recurrent. In the critical case \\(\\rho=1\\) the process is recurrent or transient according to how \\(k c(k)\\) converges to \\(1\\). When \\(\\rho=0\\) and under some regularity conditions, the support of the increments is eventually finite.

The Central Limit Theorem (CLT) for additive functionals of Markov chains is a well-known result with a long history. In this paper, we present applications to two finite-memory versions of the Elephant Random Walk, solving a problem from Gut and Stadtmüeller (2018). We also present a derivation of the CLT for additive functionals of finite state Markov chains, which is based on positive recurrence, the CLT for IID sequences and some elementary linear algebra, and which focuses on characterization of the variance.

We study a discrete-time stochastic process that can also be interpreted as a model for a viral evolution. A distinguishing feature of our process is power-law tails due to dynamics that resembles preferential attachment models. In the model we study, a population is partitioned into sites, with each site labeled by a uniquely-assigned real number in the interval \\([0,1]\\) known as fitness. The population size is a discrete-time transient birth-and-death process with probability \\(p\\) of birth and \\(1-p\\) of death. The fitness is assigned at birth according to the following rule: the new member of the population either \"mutates\" with probability \\(r\\), creating a new site uniformly distributed on \\([0,1]\\) or \"inherits\" with probability \\(1-r\\), joining an existing site with probability proportional to the site's size. At each death event, a member from the site with the lowest fitness is killed. The number of sites eventually tends to infinity if and only if \\(pr>1-p\\). Under this assumption, we show that as time tends to infinity, the joint empirical measure of site size and fitness (proportion of population in sites of size and fitness in given ranges) converges a.s. to the product of a modified Yule distribution and the uniform distribution on \\([(1-p)/(pr),1]\\). Our approach is based on the method developed in \\cite{similar-but-different}. The model and the results were independently obtained by Roy and Tanemura in [RT].