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We present an alternative view for the study of optimal control of partially observed Markov Decision Processes (POMDPs). We first revisit the traditional (and by now standard) separated-design method of reducing the problem to fully observed MDPs (belief-MDPs), and present conditions for the existence of optimal policies. Then, rather than working with this standard method, we define a Markov chain taking values in an infinite dimensional product space with the history process serving as the controlled state process and a further refinement in which the control actions and the state process are causally conditionally independent given the measurement/information process. We provide new sufficient conditions for the existence of optimal control policies under the discounted cost and average cost infinite horizon criteria. In particular, while in the belief-MDP reduction of POMDPs, weak Feller condition requirement imposes total variation continuity on either the system kernel or the measurement kernel, with the approach of this paper only weak continuity of both the transition kernel and the measurement kernel is needed (and total variation continuity is not) together with regularity conditions related to filter stability. For the discounted cost setup, we establish near optimality of finite window policies via a direct argument involving near optimality of quantized approximations for MDPs under weak Feller continuity, where finite truncations of memory can be viewed as quantizations of infinite memory with a uniform diameter in each finite window restriction under the product metric. For the average cost setup, we provide new existence conditions and also a general approach on how to initialize the randomness which we show to establish convergence to optimal cost. In the control-free case, our analysis leads to new and weak conditions for the existence and uniqueness of invariant probability measures for nonlinear filter processes, where we show that unique ergodicity of the measurement process and a measurability condition related to filter stability leads to unique ergodicity. [ABSTRACT FROM AUTHOR]

We propose a general framework for the study of the genealogy of neutral discrete-time populations. We remove the standard assumption of exchangeability of offspring distributions appearing in Cannings models, and replace it by a less restrictive condition of non-heritability of reproductive success. We provide a general criterion for the weak convergence of their genealogies to Ξ -coalescents, and apply it to a simple parametrization of our scenario (which, under mild conditions, we also prove to essentially include the general case). We provide examples for such populations, including models with highly-asymmetric offspring distributions and populations undergoing random but recurrent bottlenecks. Finally we study the limit genealogy of a new exponential model which, as previously shown for related models and in spite of its built-in (fitness) inheritance mechanism, can be brought into our setting. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a fractional analogue of the Wiener polynomial chaos expansion. It is important to highlight that the fractional order relates to the order of chaos decomposition elements, and not to the process itself, which remains the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as H α (x , y) , referred to herein as a power-normalised parabolic cylinder function. Through careful analysis of several fundamental deterministic and stochastic properties, we affirm that this function essentially serves as a fractional extension of the Hermite polynomial. In particular, the power-normalised parabolic cylinder function with the Wiener process and time as its arguments, H α (W t , t) , demonstrates martingale properties and can be interpreted as a fractional Itô integral with 1 as the integrand, thereby drawing parallels with its non-fractional counterpart. To build a fractional analogue of polynomial Wiener chaos on the real line, we introduce a new function, which we call the extended Hermite function, by smoothly joining two power-normalized parabolic cylinder functions at zero. We form an orthogonal set of extended Hermite functions as a one-parameter family and use tensor products of the extended Hermite functions as building blocks in the fractional Wiener chaos expansion, in the same way that tensor products of Hermite polynomials are used as building blocks in the Wiener chaos polynomial expansion. [ABSTRACT FROM AUTHOR]

This paper extends the results provided in Jasso-Fuentes et al. (Appl Math Optim 81(2):409–441, 2020b) and Jasso-Fuentes et al. (Pure Appl Funct Anal 9(3):675–704, 2024) regarding the study of discrete-time hybrid stochastic models with general spaces and total discounted payoffs. This extension incorporates the handling of negative and/or unbounded costs per stage. In particular, it encompasses interesting applications, such as scenarios where the controller optimizes net costs, social welfare costs, or distances between points. These situations arise when assumptions of both non-negativeness and boundedness on the cost per stage do not apply. Our proposal relies on Lyapunov-like conditions, enabling, among other aspects, the finiteness of the value function and the existence of solutions to the associated dynamic programming equation. This equation is crucial for deriving optimal control policies. To illustrate our theory, we include an example in inventory-manufacturing management, highlighting its evident hybrid nature. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic Lotka-Volterra system with infinite delay is considered. A new concept of extinction, namely, the almost sure β-extinction is proposed and sufficient conditions for the solution to be almost sure β-extinction are obtained. When the positive equilibrium exists and the intensities of the noises are small enough, any solution of the system is attracted by the positive equilibrium. Finally, numerical simulations are carried out to support the results. [ABSTRACT FROM AUTHOR]

In this paper, random environment and random time delay are introduced into general nonlinear autoregressive models to improve the applicability of these models. The introduction of random environment makes nonlinear autoregressive models more flexible and more widely used. Random time delay is a new way for models to be applied in new fields. In order to analyze the properties of the proposed new model, we construct a new stochastic process and prove that it is an irreducible and aperiodic time-homogeneous Markov chain. The nonrecurrence of the proposed new type of models is considered, and some sufficient conditions for nonrecurrence are investigated. [ABSTRACT FROM AUTHOR]

In this work, we characterize the fixed points of the doubly stochastic operators defined on discrete spaces $ \ell ^p(I) , $ ℓ p (I) , where I is a nonempty set and $ p\in [1,+\infty). $ p ∈ [ 1 , + ∞). We use our results to characterize all Markov chains with infinite state space and doubly stochastic transition probability matrix that have at least one stationary distribution vector. [ABSTRACT FROM AUTHOR]

We give rates of convergence in the almost sure invariance principle for sums of dependent random variables with semi-exponential tails, whose coupling coefficients decrease at a sub-exponential rate. We show that the rates in the strong invariance principle are in powers of log n . We apply our results to iid products of random matrices. [ABSTRACT FROM AUTHOR]

This paper investigates the asymptotic analysis of the hitting time of Markov-type jump processes (i.e., semi-Markov, Markov, in continuous or discrete time) with a small probability of entering a non-empty terminal subset. This means that absorption is a rare event. The mean hitting time function of all four type processes obeyed the same equation. We obtain unified results of asymptotic approximation in a series scheme or, equivalently, a functional type of mean hitting time. [ABSTRACT FROM AUTHOR]

This paper studies constrained Markov decision processes under the total expected discounted cost optimality criterion, with a state-action dependent discount factor that may take any value between zero and one. Both the state and the action space are assumed to be Borel spaces. By using the linear programming approach, consisting in stating the control problem as a linear problem on a set of occupation measures, we show the existence of an optimal stationary Markov policy. Our results are based on the study of both weak-strong topologies in the space of occupation measures and Young measures in the space of Markov policies. [ABSTRACT FROM AUTHOR]

In this paper, we consider a subclass of piecewise deterministic Markov processes with a Polish state space that involve a deterministic motion punctuated by random jumps, occurring in a Poisson-like fashion with some state-dependent rate, between which the trajectory is driven by one of the given semiflows. We prove that there is a one-to-one correspondence between stationary distributions of such processes and those of the Markov chains given by their post-jump locations. Using this result, we further establish a criterion guaranteeing the existence and uniqueness of the stationary distribution in a particular case, where the post-jump locations result from the action of a random iterated function system with an arbitrary set of transformations. [ABSTRACT FROM AUTHOR]

This article studies deep neural network expression rates for optimal stopping problems of discrete-time Markov processes on high-dimensional state spaces. A general framework is established in which the value function and continuation value of an optimal stopping problem can be approximated with error at most ε by a deep ReLU neural network of size at most κ d q ε − r . The constants κ , q , r ≥ 0 do not depend on the dimension d of the state space or the approximation accuracy ε . This proves that deep neural networks do not suffer from the curse of dimensionality when employed to approximate solutions of optimal stopping problems. The framework covers for example exponential Lévy models, discrete diffusion processes and their running minima and maxima. These results mathematically justify the use of deep neural networks for numerically solving optimal stopping problems and pricing American options in high dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we prove the following assertion for an absorbing Markov decision process (MDP) with the given initial distribution, which is also assumed to be semi-continuous: the continuity of the projection mapping from the space of strategic measures to the space of occupation measures, both endowed with their weak topologies, is equivalent to the MDP model being uniformly absorbing. An example demonstrates, among other interesting scenarios, that for an absorbing (but not uniformly absorbing) semi-continuous MDP with the given initial distribution, the space of occupation measures can fail to be compact in the weak topology. [ABSTRACT FROM AUTHOR]

A Hidden Markov Model (HMM) is a couple of stochastic processes: A hidden Markov process and an observed emission process. Generally, the HMMs are used to study the hidden behavior of random systems through some observed emission sequences generated by the phenomenon under study. In this frame-work, we propose to solve the likelihood and the decoding problems of HMMs whose state space is composed of a continuous part and a discrete part. We adapt forward, backward and Viterbi algorithms to the case of our proposal. Numerical examples and Monte Carlo simulations are considered to show the efficiency and the adaptation of the algorithms for the proposed model. [ABSTRACT FROM AUTHOR]

In this paper, we consider (upward skip-free) discrete-time and discrete-space Markov additive chains (MACs) and develop the theory for the so-called W ~ and Z ~ scale matrices, which are shown to play a vital role in the determination of a number of exit problems and related fluctuation identities. The theory developed in this fully discrete set-up follows similar lines of reasoning as the analogous theory for Markov additive processes in continuous time and is exploited to obtain the probabilistic construction of the scale matrices, identify the form of the generating function and produce a simple recursion relation for W ~ , as well as its connection with the so-called occupation mass formula. In addition to the standard one- and two-sided exit problems (upwards and downwards), we also derive distributional characteristics for a number of quantities related to the one- and two-sided 'reflected' processes. [ABSTRACT FROM AUTHOR]

The transition kernel of an ℝn-valued diffusion or jump diffusion process {Xt} is known to satisfy the Feller property if {Xt} is the solution of an SDE whose coefficients are Lipschitz continuous. This Lipschitz route to Feller falls short if {Xt} is the solution of an SDE whose coefficients depend on a state-dependent regime-switching process {θt}. In this paper it is shown that pathwise uniqueness and the Feller property are satisfied under mild conditions for a regime-switching jump diffusion process {Xt, θt} with hybrid jumps, i.e. jumps in {Xt} that occur simultaneously with {θt} switching. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions for the almost sure central limit theorem started at a point, known under the name of quenched central limit theorem. This is achieved by using a new idea of conditioning with respect to both the past and the future of the Markov chain. As applications, we provide a new sufficient projective condition for the quenched CLT. [ABSTRACT FROM AUTHOR]

Let (gn)n⩾l be a sequence of independent and identically distributed positive random d × d matrices and consider the matrix product Gn = gn⋯ g1. Under suitable conditions, we establish the Berry-Esseen bounds on the rate of convergence in the central limit theorem and Cramér-type moderate deviation expansions, for any matrix norm ∥Gn∥ of Gn, its entries G n i , j and its spectral radius ρ(Gn). Extended versions of their joint law with the direction of the random walk Gnx are also established, where x is a starting point in the unit sphere of ℝd. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic behaviour of the sequence of forward type iterations of a given random-valued vector function on the state space being a separable and complete metric space. Assuming non-linear contraction in mean we prove that the considered sequence converges weakly to a random variable with a finite first moment and independent of the initial state. Moreover, we show that the speed of this convergence does not have to be geometric. We also present examples illustrating the result obtained. [ABSTRACT FROM AUTHOR]

Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Among others, we study cases where the interarrival and service times are proportionally dependent on additive and/or subtracting delay, as well as cases where interarrival times depend on whether the service duration of the previous arrival exceeds or not a random threshold. Moreover, we study cases where the autoregressive parameter is constant as well as a discrete or a continuous random variable. More general dependence structures are also discussed. Our primary aim is to investigate a broad class of recursions of autoregressive type for which several independence assumptions are lifted and for which a detailed exact analysis is provided. We provide expressions for the Laplace transform of the waiting time distribution of a customer in the system in terms of an infinite sum of products of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with impatient customers and a general dependence structure is also considered. For such a model, we provide expressions for the probability generating function of the stationary orbit queue length distribution in terms of an infinite sum of products of known generating functions. A first attempt towards a multidimensional setting is also considered. [ABSTRACT FROM AUTHOR]

Polar slice sampling, a Markov chain construction for approximate sampling, performs, under suitable assumptions on the target and initial distribution, provably independent of the state space dimension. We extend the aforementioned result of Roberts and Rosenthal (Stoch Model 18(2):257–280, 2002) by developing a theory which identifies conditions, in terms of a generalized level set function, that imply an explicit lower bound on the spectral gap even in a general slice sampling context. Verifying the identified conditions for polar slice sampling yields a lower bound of 1/2 on the spectral gap for arbitrary dimension if the target density is rotationally invariant, log-concave along rays emanating from the origin and sufficiently smooth. The general theoretical result is potentially applicable beyond the polar slice sampling framework. [ABSTRACT FROM AUTHOR]

In this study, we consider disorder detection problems for statistical models with dependent observations defined by Markov chains in a Bayesian setting for a uniform prior distribution when the number of observations is limited by some known quantity. To achieve this, a new optimal sequential detection procedure is constructed using optimal stopping methods. This procedure minimizes the average delay time in the class of sequential procedures with false alarm probabilities that do not exceed a fixed value. The main difference between the proposed detection procedure and the conventional ones is that it is based not on the posterior probabilities but on the weighted Shiryaev-Roberts statistic. This allows for optimal detection in a nonasymptotic sense over any bounded time interval. Thereafter, we applied the constructed procedures to the early detection problem for the beginning of the epidemic spread. We used two epidemic models: the binomial model proposed by Baron, Choudhary, and Yu (2013) and the model based on the Gaussian approximation introduced by Pergamenchtchikov, Tartakovsky, and Spivak (2022). The theoretical results were confirmed through numerical simulations using the Monte Carlo method. [ABSTRACT FROM AUTHOR]

We study a natural random walk on the n × n uni-upper triangular matrices, with entries in Z / m Z , generated by steps which add or subtract a uniformly random row to the row above. We show that the mixing time of this random walk is O (m 2 n log n + n 2 m o (1)) . This answers a question of Stong and of Arias-Castro, Diaconis, and Stanley. [ABSTRACT FROM AUTHOR]

We study the ergodic behaviour of a discrete-time process X which is a Markov chain in a stationary random environment. The laws of X t are shown to converge to a limiting law in (weighted) total variation distance as t → ∞ . Convergence speed is estimated, and an ergodic theorem is established for functionals of X. Our hypotheses on X combine the standard "drift" and "small set" conditions for geometrically ergodic Markov chains with conditions on the growth rate of a certain "maximal process" of the random environment. We are able to cover a wide range of models that have heretofore been intractable. In particular, our results are pertinent to difference equations modulated by a stationary (Gaussian) process. Such equations arise in applications such as discretized stochastic volatility models of mathematical finance. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of the finite-dimensional distributions and asymptotic behavior of extremal Markov processes connected with the Kendall convolution. In particular, we provide general formulas for the finite dimensional distributions of the random walk driven by the Kendall convolution for a large class of step size distributions. Moreover, we prove limit theorems for random walks and associated continuous-time stochastic processes. [ABSTRACT FROM AUTHOR]

The paper is concerned with a new approach for the recurrence properties of the oscillating random walk on Z in the sense of Kemperman. In the case when the random walk is ascending on Z - and descending on Z + , the invariant measure of the embedded process of successive crossing times is explicitly determined, which yields a sufficient condition for recurrence. Finally, we make use of this result to show that the general oscillating random walk is recurrent under some moment assumptions. [ABSTRACT FROM AUTHOR]

We consider a family of Markov chains whose transition dynamics are affected by model parameters. Understanding the parametric dependence of (complex) performance measures of such Markov chains is often of significant interest. The derivatives and their continuity of the performance measures w.r.t. the parameters play important roles, for example, in numerical optimization of the performance measures, and quantification of the uncertainties in the performance measures when there are uncertainties in the parameters from the statistical estimation procedures. In this paper, we establish conditions that guarantee the smoothness of various types of intractable performance measures—such as the stationary and random horizon discounted performance measures—of general state space Markov chains and provide probabilistic representations for the derivatives. Funding: C.-H. Rhee is supported by the National Science Foundation [Grant CMMI-2146530]. [ABSTRACT FROM AUTHOR]

Gibbs sampling, also known as Coordinate Hit-and-Run (CHAR), is a Markov chain Monte Carlo algorithm for sampling from high-dimensional distributions. In each step, the algorithm selects a random coordinate and re-samples that coordinate from the distribution induced by fixing all the other coordinates. While this algorithm has become widely used over the past half-century, guarantees of efficient convergence have been elusive. We show that the Coordinate Hit-and-Run algorithm for sampling from a convex body K in R n mixes in O ∗ (n 9 R 2 / r 2) steps, where K contains a ball of radius r and R is the average distance of a point of K from its centroid. We also give an upper bound on the conductance of Coordinate Hit-and-Run, showing that it is strictly worse than Hit-and-Run or the Ball Walk in the worst case. [ABSTRACT FROM AUTHOR]

In this article, an almost sure central limit theorem for non-stationary Markov chains on a general state space is established by using the techniques of shift operator and harmonic function. [ABSTRACT FROM AUTHOR]

This work concerns with Markov decision processes on a denumerable state space. It is assumed that the performance of a control policy is measured by the average criterion associated with a risk-seeking controller with constant risk-sensitivity coefficient. The structural assumptions on the model ensure that the optimal average cost is constant, but it is possible that the optimalty equation does not admit a solution. In this context, a risk-sensitive version of the classical discounted approach is used to obtain convergent approximations to the optimal average cost, and to determine nearly optimal stationary policies. [ABSTRACT FROM AUTHOR]

We consider a policy gradient algorithm applied to a finite-arm bandit problem with Bernoulli rewards. We allow learning rates to depend on the current state of the algorithm rather than using a deterministic time-decreasing learning rate. The state of the algorithm forms a Markov chain on the probability simplex. We apply Foster–Lyapunov techniques to analyze the stability of this Markov chain. We prove that, if learning rates are well-chosen, then the policy gradient algorithm is a transient Markov chain, and the state of the chain converges on the optimal arm with logarithmic or polylogarithmic regret. [ABSTRACT FROM AUTHOR]

In this article we propose a new online control system aiming to lower the instants in which the production process migrates from being in- to out-of-control state, which generates an increase in the non-conformity rates. As shifts from in- to out-of-control are non-deterministic, a sample of size n is collected, for each m or L units produced, and each element from the sample is imprecisely classified as conform or non-conform (that is, there may be classification errors). If the amount of conform units from the sample is equal or greater than a, the process would not be adjusted and another sample would be collected after m units produced. If the quantity of conform units is inferior to a, the process would be adjusted and another sample would be collected after L units produced, given that L > m. A genetic algorithm is proposed to approximately find the values of a, n, m, and L that minimize all costs involved in the process being controlled. All procedures are illustrated through a detailed numerical example that attests the efficacy and efficiency of the proposed online control system. [ABSTRACT FROM AUTHOR]

We show that an invariant measure of a Markov operator that is contracting in the Hutchison distance, acting on a space of Borel probability measures on a Polish space, depends continuously on the given operator. In addition, we establish an estimate for a distance between invariant measures. Some applications to the weak limit of iterates of random-valued functions (in particular, the so-called random affine maps, occuring, e.g., in perpetuities analysis) are also given. [ABSTRACT FROM AUTHOR]

Our main goal in this paper is to propose a mathematical model in order to use it to study the historical achievements of a particular team in order to stand on the evolutionary behavior of the team and to be able to predict the future results of the team by measuring the team's performance through the total number of victories, losses and draws in official matches approved. Our main idea is to use a Markov model to find the results of matches, so that the outcome of each match is one of the specified cases: win or lose or draw. [ABSTRACT FROM AUTHOR]

Order statistics arising from 푚 independent but not identically distributed random variables are typically constructed by arranging some X 1 , X 2 , ... , X m , with X i having distribution function F i ⁢ (x) , in increasing order denoted as X (1) ≤ X (2) ≤ ⋯ ≤ X (m) . In this case, X (i) is not necessarily associated with F i ⁢ (x) . Assuming one can simulate values from each distribution, one can generate such "non-iid" order statistics by simulating X i from F i , for i = 1 , 2 , ... , m , and arranging them in order. In this paper, we consider the problem of simulating ordered values X (1) , X (2) , ... , X (m) such that the marginal distribution of X (i) is F i ⁢ (x) . This problem arises in Bayesian principal components analysis (BPCA) where the X i are ordered eigenvalues that are a posteriori independent but not identically distributed. We propose a novel coupling-from-the-past algorithm to "perfectly" (up to computable order of accuracy) simulate such order-constrained non-iid order statistics. We demonstrate the effectiveness of our approach for several examples, including the BPCA problem. [ABSTRACT FROM AUTHOR]

Consider a discrete-time homogeneous Markov chain with initial state i. We study the distribution of L (j , n) , the length of the longest consecutive visits of this chain to state j until time n. We provide two limiting theorems for L (j , n) and establish asymptotics for the moment generating function of L (j , n). We conclude by closing the open problem raised by the authors of [T. Konstantopoulos, Z. Liu, and X. Yang, Laplace transform asymptotics and large deviation principles for longest success runs in Bernoulli trials, J. Appl. Probab. 53 (2016), pp. 747–764.] by providing two large deviation principles for L (j , n). [ABSTRACT FROM AUTHOR]

Bifurcating Markov chains (BMCs) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for additive functionals of BMCs under $L^2$ -ergodic conditions with three different regimes. This completes the pointwise approach developed in a previous work. As an application, we study the elementary case of a symmetric bifurcating autoregressive process, which justifies the nontrivial hypothesis considered on the kernel transition of the BMCs. We illustrate in this example the phase transition observed in the fluctuations. [ABSTRACT FROM AUTHOR]

We introduce a new definition of bubbles in discrete-time models based on the discounted stock price losing mass under an equivalent martingale measure at some finite drawdown. We provide equivalent probabilistic characterisations of this definition and give examples of discrete-time martingales that are bubbles and others that are not. In the Markovian case, we provide sufficient analytic conditions for the presence of bubbles. We also show that the existence of bubbles is directly linked to the existence of a non-trivial solution to a linear Volterra integral equation of the second kind involving the Markov kernel. Finally, we show that our definition of bubbles in discrete time is consistent with the strict local martingale definition of bubbles in continuous time in the sense that a properly discretised strict local martingale in continuous time is a bubble in discrete time. [ABSTRACT FROM AUTHOR]

Markov chain Monte Carlo (MCMC) methods are important in a variety of statistical applications that require sampling from intractable probability distributions. Among the most common MCMC algorithms is the Gibbs sampler. When an MCMC algorithm is used, it is important to have an idea of how long it takes for the chain to become "close" to its stationary distribution. In many cases, there is high autocorrelation in the output of the chain, so the output needs to be thinned so that an approximate random sample from the desired probability distribution can be obtained by taking a state of the chain every h steps in a process called h-thinning. This manuscript extends the work of [D. A. Spade, Estimating drift and minorization coefficients for Gibbs sampling algorithms, Monte Carlo Methods Appl. 27 2021, 3, 195–209] by presenting a computational approach to obtaining an approximate upper bound on the mixing time of the h-thinned Gibbs sampler. [ABSTRACT FROM AUTHOR]

We present new criteria for geometric and algebraic transience for discrete-time transient Markov chains on general state spaces, based on the moment of the last exit time, the modified moment of the first return time and the drift condition for the transition kernel. These criteria turn out to be more convenient to use, supplementing and extending conditions introduced by Mao and Song [Stochastic Process. Appl. 124 (2014) 1648-1678]. Several applications are presented including discrete queueing Markov chains, Galton–Watson branching processes, downwardly skip-free chains, unrestricted random walks and autoregressive models of order one. [ABSTRACT FROM AUTHOR]

Motivated by recent studies of big samples, this work aims to construct a parametric model which is characterized by the following features: (i) a 'local' reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term almost sure convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness-of-fit result for the limit probabilities. This triple purpose is achieved by the introduction of a new variant of the Eggenberger–Pólya urn, which we call the rescaled Pólya urn. We provide a complete asymptotic characterization of this model, pointing out that, for a certain choice of the parameters, it has properties different from the ones typically exhibited by the other urn models in the literature. Therefore, beyond the possible statistical application, this work could be interesting for those who are concerned with stochastic processes with reinforcement. [ABSTRACT FROM AUTHOR]

Simulated tempering is a popular method of allowing Markov chain Monte Carlo algorithms to move between modes of a multimodal target density $\pi$. Tawn, Moores and Roberts (2021) introduces the Annealed Leap-Point Sampler (ALPS) to allow for rapid movement between modes. In this paper we prove that, under appropriate assumptions, a suitably scaled version of the ALPS algorithm converges weakly to skew Brownian motion. Our results show that, under appropriate assumptions, the ALPS algorithm mixes in time $O(d [\log d]^2)$ or O(d), depending on which version is used. [ABSTRACT FROM AUTHOR]