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Artículos de revistas sobre el tema "Convergence of Markov processes"

Autor: Grafiati

Publicado: 23 de noviembre de 2024

Última modificación: 31 de julio de 2025

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1

Abakuks, A., S. N. Ethier, and T. G. Kurtz. "Markov Processes: Characterization and Convergence." Biometrics 43, no. 2 (1987): 484. http://dx.doi.org/10.2307/2531839.

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2

Perkins, Edwin, S. N. Ethier, and T. G. Kurtz. "Markov Processes, Characterization and Convergence." Journal of the Royal Statistical Society. Series A (Statistics in Society) 151, no. 2 (1988): 367. http://dx.doi.org/10.2307/2982773.

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3

Franz, Uwe, Volkmar Liebscher, and Stefan Zeiser. "Piecewise-Deterministic Markov Processes as Limits of Markov Jump Processes." Advances in Applied Probability 44, no. 3 (2012): 729–48. http://dx.doi.org/10.1239/aap/1346955262.

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A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes ap

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4

Franz, Uwe, Volkmar Liebscher, and Stefan Zeiser. "Piecewise-Deterministic Markov Processes as Limits of Markov Jump Processes." Advances in Applied Probability 44, no. 03 (2012): 729–48. http://dx.doi.org/10.1017/s0001867800005851.

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A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes ap

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5

Swishchuk, Anatoliy, and M. Shafiqul Islam. "Diffusion Approximations of the Geometric Markov Renewal Processes and Option Price Formulas." International Journal of Stochastic Analysis 2010 (December 19, 2010): 1–21. http://dx.doi.org/10.1155/2010/347105.

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We consider the geometric Markov renewal processes as a model for a security market and study this processes in a diffusion approximation scheme. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes in diffusion scheme are presented. We present European call option pricing formulas in the case of ergodic, double-averaged, and merged diffusion geometric Markov renewal processes.

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6

Aldous, David J. "Book Review: Markov processes: Characterization and convergence." Bulletin of the American Mathematical Society 16, no. 2 (1987): 315–19. http://dx.doi.org/10.1090/s0273-0979-1987-15533-9.

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7

Crank, Keith N., and Prem S. Puri. "A method of approximating Markov jump processes." Advances in Applied Probability 20, no. 1 (1988): 33–58. http://dx.doi.org/10.2307/1427269.

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We present a method of approximating Markov jump processes which was used by Fuhrmann [7] in a special case. We generalize the method and prove weak convergence results under mild assumptions. In addition we obtain bounds on the rates of convergence of the probabilities at arbitrary fixed times. The technique is demonstrated using a state-dependent branching process as an example.

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8

Crank, Keith N., and Prem S. Puri. "A method of approximating Markov jump processes." Advances in Applied Probability 20, no. 01 (1988): 33–58. http://dx.doi.org/10.1017/s0001867800017936.

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We present a method of approximating Markov jump processes which was used by Fuhrmann [7] in a special case. We generalize the method and prove weak convergence results under mild assumptions. In addition we obtain bounds on the rates of convergence of the probabilities at arbitrary fixed times. The technique is demonstrated using a state-dependent branching process as an example.

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9

Deng, Chang-Song, René L. Schilling, and Yan-Hong Song. "Subgeometric rates of convergence for Markov processes under subordination." Advances in Applied Probability 49, no. 1 (2017): 162–81. http://dx.doi.org/10.1017/apr.2016.83.

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Abstract We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent), we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moment (subexponential, algebraic, and logarithmic) for subordinators as time t tends to ∞. We also discuss some concrete models and we show that subo

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10

Lam, Hoang-Chuong. "Weak Convergence for Markov Processes on Discrete State Spaces." Markov Processes And Related Fields, no. 2024 № 4 (30) (February 8, 2025): 587–98. https://doi.org/10.61102/1024-2953-mprf.2024.30.4.004.

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This study investigates the weak convergence of Markov processes on discrete state spaces under the assumption that the transition intensities converge to a constant. Additionally, the research determines the limits of higher-order moments of the Markov process, which are utilized to prove the existence of limit theorems.

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11

Malyk, Igor V. "Compensating Operator and Weak Convergence of Semi-Markov Process to the Diffusion Process without Balance Condition." Journal of Applied Mathematics 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/563060.

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Weak convergence of semi-Markov processes in the diffusive approximation scheme is studied in the paper. This problem is not new and it is studied in many papers, using convergence of random processes. Unlike other studies, we used in this paper concept of the compensating operator. It enables getting sufficient conditions of weak convergence under the conditions on the local characteristics of output semi-Markov process.

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12

Swishchuk, Anatoliy, and Nikolaos Limnios. "Controlled Discrete-Time Semi-Markov Random Evolutions and Their Applications." Mathematics 9, no. 2 (2021): 158. http://dx.doi.org/10.3390/math9020158.

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In this paper, we introduced controlled discrete-time semi-Markov random evolutions. These processes are random evolutions of discrete-time semi-Markov processes where we consider a control. applied to the values of random evolution. The main results concern time-rescaled weak convergence limit theorems in a Banach space of the above stochastic systems as averaging and diffusion approximation. The applications are given to the controlled additive functionals, controlled geometric Markov renewal processes, and controlled dynamical systems. We provide dynamical principles for discrete-time dynam

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13

Champagnat, Nicolas, and Denis Villemonais. "Uniform convergence of penalized time-inhomogeneous Markov processes." ESAIM: Probability and Statistics 22 (2018): 129–62. http://dx.doi.org/10.1051/ps/2017022.

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We provide a general criterion ensuring the exponential contraction of Feynman–Kac semi-groups of penalized processes. This criterion applies to time-inhomogeneous Markov processes with absorption and killing through penalization. We also give the asymptotic behavior of the expected penalization and provide results of convergence in total variation of the process penalized up to infinite time. For exponential convergence of penalized semi-groups with bounded penalization, a converse result is obtained, showing that our criterion is sharp in this case. Several cases are studied: we first show h

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14

Xia, Aihua. "Weak Convergence of Markov Processes with Extended Generators." Annals of Probability 22, no. 4 (1994): 2183–202. http://dx.doi.org/10.1214/aop/1176988499.

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15

Mao, Yong-Hua. "Convergence rates in strong ergodicity for Markov processes." Stochastic Processes and their Applications 116, no. 12 (2006): 1964–76. http://dx.doi.org/10.1016/j.spa.2006.05.008.

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16

Turner, Amanda G. "Convergence of Markov processes near saddle fixed points." Annals of Probability 35, no. 3 (2007): 1141–71. http://dx.doi.org/10.1214/009117906000000836.

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17

Liu, Yuanyuan, and Zhenting Hou. "Several Types of Ergodicity for M/G/1-Type Markov Chains and Markov Processes." Journal of Applied Probability 43, no. 1 (2006): 141–58. http://dx.doi.org/10.1239/jap/1143936249.

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In this paper we study polynomial and geometric (exponential) ergodicity for M/G/1-type Markov chains and Markov processes. First, practical criteria for M/G/1-type Markov chains are obtained by analyzing the generating function of the first return probability to level 0. Then the corresponding criteria for M/G/1-type Markov processes are given, using their h-approximation chains. Our method yields the radius of convergence of the generating function of the first return probability, which is very important in obtaining explicit bounds on geometric (exponential) convergence rates. Our results a

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18

Liu, Yuanyuan, and Zhenting Hou. "Several Types of Ergodicity for M/G/1-Type Markov Chains and Markov Processes." Journal of Applied Probability 43, no. 01 (2006): 141–58. http://dx.doi.org/10.1017/s002190020000142x.

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In this paper we study polynomial and geometric (exponential) ergodicity for M/G/1-type Markov chains and Markov processes. First, practical criteria for M/G/1-type Markov chains are obtained by analyzing the generating function of the first return probability to level 0. Then the corresponding criteria for M/G/1-type Markov processes are given, using their h-approximation chains. Our method yields the radius of convergence of the generating function of the first return probability, which is very important in obtaining explicit bounds on geometric (exponential) convergence rates. Our results a

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19

Macci, Claudio. "Continuous-time Markov additive processes: Composition of large deviations principles and comparison between exponential rates of convergence." Journal of Applied Probability 38, no. 4 (2001): 917–31. http://dx.doi.org/10.1239/jap/1011994182.

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We consider a continuous-time Markov additive process (Jt,St) with (Jt) an irreducible Markov chain on E = {1,…,s}; it is known that (St/t) satisfies the large deviations principle as t → ∞. In this paper we present a variational formula H for the rate function κ∗ and, in some sense, we have a composition of two large deviations principles. Moreover, under suitable hypotheses, we can consider two other continuous-time Markov additive processes derived from (Jt,St): the averaged parameters model (Jt,St(A)) and the fluid model (Jt,St(F)). Then some results of convergence are presented and the va

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20

Cipra, Tomáš. "Autoregressive processes in optimization." Journal of Applied Probability 25, no. 2 (1988): 302–12. http://dx.doi.org/10.2307/3214438.

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Vector autoregressive processes of the first order are considered which are non-negative and optimize a linear objective function. These processes may be used in stochastic linear programming with a dynamic structure. By using Tweedie's results from the theory of Markov chains, conditions for geometric rates of convergence to stationarity (i.e. so-called geometric ergodicity) and for existence and geometric convergence of moments of these processes are obtained.

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21

Cipra, Tomáš. "Autoregressive processes in optimization." Journal of Applied Probability 25, no. 02 (1988): 302–12. http://dx.doi.org/10.1017/s0021900200040948.

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Vector autoregressive processes of the first order are considered which are non-negative and optimize a linear objective function. These processes may be used in stochastic linear programming with a dynamic structure. By using Tweedie's results from the theory of Markov chains, conditions for geometric rates of convergence to stationarity (i.e. so-called geometric ergodicity) and for existence and geometric convergence of moments of these processes are obtained.

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22

Haas, Peter J., and Gerald S. Shedler. "Stochastic Petri Nets: Modeling Power and Limit Theorems." Probability in the Engineering and Informational Sciences 5, no. 4 (1991): 477–98. http://dx.doi.org/10.1017/s0269964800002242.

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Generalized semi-Markov processes and stochastic Petri nets provide building blocks for specification of discrete event system simulations on a finite or countable state space. The two formal systems differ, however, in the event scheduling (clock-setting) mechanism, the state transition mechanism, and the form of the state space. We have shown previously that stochastic Petri nets have at least the modeling power of generalized semi-Markov processes. In this paper we show that stochastic Petri nets and generalized semi-Markov processes, in fact, have the same modeling power. Combining this re

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23

Macci, Claudio. "Continuous-time Markov additive processes: Composition of large deviations principles and comparison between exponential rates of convergence." Journal of Applied Probability 38, no. 04 (2001): 917–31. http://dx.doi.org/10.1017/s0021900200019136.

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We consider a continuous-time Markov additive process (J t ,S t ) with (J t ) an irreducible Markov chain on E = {1,…,s}; it is known that (S t /t) satisfies the large deviations principle as t → ∞. In this paper we present a variational formula H for the rate function κ∗ and, in some sense, we have a composition of two large deviations principles. Moreover, under suitable hypotheses, we can consider two other continuous-time Markov additive processes derived from (J t ,S t ): the averaged parameters model (J t ,S t (A)) and the fluid model (J t ,S t (F)). Then some results of convergence are

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24

Kalpazidou, Sophia. "On the weak convergence of sequences of circuit processes: a probabilistic approach." Journal of Applied Probability 29, no. 2 (1992): 374–83. http://dx.doi.org/10.2307/3214574.

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The asymptotic behaviour of sequences of Markov processes whose finite distributions depend upon the sample paths ω of a positive recurrent Markov chain ξ is studied. The existence of such sequences depends upon the existence of a unique class of directed weighted circuits having a probabilistic interpretation in terms of the directed circuits occurring along the sample paths of ξ. An application to multiple Markov chains is given.

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25

Kalpazidou, Sophia. "On the weak convergence of sequences of circuit processes: a probabilistic approach." Journal of Applied Probability 29, no. 02 (1992): 374–83. http://dx.doi.org/10.1017/s0021900200043126.

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The asymptotic behaviour of sequences of Markov processes whose finite distributions depend upon the sample paths ω of a positive recurrent Markov chain ξ is studied. The existence of such sequences depends upon the existence of a unique class of directed weighted circuits having a probabilistic interpretation in terms of the directed circuits occurring along the sample paths of ξ. An application to multiple Markov chains is given.

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26

HWANG, CHII-RUEY. "ACCELERATING MONTE CARLO MARKOV PROCESSES." COSMOS 01, no. 01 (2005): 87–94. http://dx.doi.org/10.1142/s0219607705000085.

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Let π be a probability density proportional to exp - U(x) in S. A convergent Markov process to π(x) may be regarded as a "conceptual" algorithm. Assume that S is a finite set. Let X0,X1,…,Xn,… be a Markov chain with transition matrix P and invariant probability π. Under suitable condition on P, it is known that [Formula: see text] converges to π(f) and the corresponding asymptotic variance v(f, P) depends only on f and P. It is natural to consider criteria vw(P) and va(P), defined respectively by maximizing and averaging v(f, P) over f. Two families of transition matrices are considered. There

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27

Cooper, William L., Shane G. Henderson, and Mark E. Lewis. "CONVERGENCE OF SIMULATION-BASED POLICY ITERATION." Probability in the Engineering and Informational Sciences 17, no. 2 (2003): 213–34. http://dx.doi.org/10.1017/s0269964803172051.

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Simulation-based policy iteration (SBPI) is a modification of the policy iteration algorithm for computing optimal policies for Markov decision processes. At each iteration, rather than solving the average evaluation equations, SBPI employs simulation to estimate a solution to these equations. For recurrent average-reward Markov decision processes with finite state and action spaces, we provide easily verifiable conditions that ensure that simulation-based policy iteration almost-surely eventually never leaves the set of optimal decision rules. We analyze three simulation estimators for soluti

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28

Qocayeva, N. "THE MARKOV THEOREM AND THE LAW OF LARGE NUMBERS: FUNDAMENTAL PRINCIPLES AND DERIVED RESULTS." Znanstvena misel journal, no. 93 (August 30, 2024): 21–24. https://doi.org/10.5281/zenodo.13582822.

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This article explores the fundamental principles and applications of the Law of Large Numbers (LLN) and the Markov Theorem. The LLN establishes the convergence of sample averages to expected values, while the Markov Theorem ensures the convergence of Markov Chains to stationary distributions. We discuss the theoretical underpinnings, practical applications in various fields, and future research directions, highlighting the importance of these concepts in understanding stochastic processes and making predictions.

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29

Szehr, Oleg, David Reeb, and Michael M. Wolf. "Spectral Convergence Bounds for Classical and Quantum Markov Processes." Communications in Mathematical Physics 333, no. 2 (2014): 565–95. http://dx.doi.org/10.1007/s00220-014-2188-5.

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30

Lund, Robert B., Sean P. Meyn, and Richard L. Tweedie. "Computable exponential convergence rates for stochastically ordered Markov processes." Annals of Applied Probability 6, no. 1 (1996): 218–37. http://dx.doi.org/10.1214/aoap/1034968072.

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31

Park, Y. S., J. C. Bean, and R. L. Smith. "Optimal Average Value Convergence in Nonhomogeneous Markov Decision Processes." Journal of Mathematical Analysis and Applications 179, no. 2 (1993): 525–36. http://dx.doi.org/10.1006/jmaa.1993.1367.

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32

Gravereaux, Jean-Bernard, and James Ledoux. "Poisson approximation for some point processes in reliability." Advances in Applied Probability 36, no. 2 (2004): 455–70. http://dx.doi.org/10.1239/aap/1086957581.

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In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

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33

Gravereaux, Jean-Bernard, and James Ledoux. "Poisson approximation for some point processes in reliability." Advances in Applied Probability 36, no. 02 (2004): 455–70. http://dx.doi.org/10.1017/s0001867800013562.

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In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

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34

LIU, YUANYUAN, HANJUN ZHANG, and YIQIANG ZHAO. "COMPUTABLE STRONGLY ERGODIC RATES OF CONVERGENCE FOR CONTINUOUS-TIME MARKOV CHAINS." ANZIAM Journal 49, no. 4 (2008): 463–78. http://dx.doi.org/10.1017/s1446181108000114.

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AbstractIn this paper, we investigate computable lower bounds for the best strongly ergodic rate of convergence of the transient probability distribution to the stationary distribution for stochastically monotone continuous-time Markov chains and reversible continuous-time Markov chains, using a drift function and the expectation of the first hitting time on some state. We apply these results to birth–death processes, branching processes and population processes.

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35

Böttcher, Björn. "Embedded Markov chain approximations in Skorokhod topologies." Probability and Mathematical Statistics 39, no. 2 (2019): 259–77. http://dx.doi.org/10.19195/0208-4147.39.2.2.

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We prove a J1-tightness condition for embedded Markov chains and discuss four Skorokhod topologies in a unified manner. To approximate a continuous time stochastic process by discrete time Markov chains, one has several options to embed the Markov chains into continuous time processes. On the one hand, there is a Markov embedding which uses exponential waiting times. On the other hand, each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for J1, the linear interpolation embedding forM1, the multistep embedding for J2 and a more general embedding

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36

Ying, Donghao, Mengzi Amy Guo, Yuhao Ding, Javad Lavaei, and Zuo-Jun Shen. "Policy-Based Primal-Dual Methods for Convex Constrained Markov Decision Processes." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 9 (2023): 10963–71. http://dx.doi.org/10.1609/aaai.v37i9.26299.

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We study convex Constrained Markov Decision Processes (CMDPs) in which the objective is concave and the constraints are convex in the state-action occupancy measure. We propose a policy-based primal-dual algorithm that updates the primal variable via policy gradient ascent and updates the dual variable via projected sub-gradient descent. Despite the loss of additivity structure and the nonconvex nature, we establish the global convergence of the proposed algorithm by leveraging a hidden convexity in the problem, and prove the O(T^-1/3) convergence rate in terms of both optimality gap and const

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37

Giroux, Gaston. "Asymptotic results for non-linear processes of the McKean tagged-molecule type." Journal of Applied Probability 23, no. 1 (1986): 42–51. http://dx.doi.org/10.2307/3214115.

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McKean's tagged-molecule process is a non-linear homogeneous two-state Markov chain in continuous time, constructed with the aid of a binary branching process. For each of a large class of branching processes we construct a similar process. The construction is carefully done and the weak homogeneity is deduced. A simple probability argument permits us to show convergence to the equidistribution (½, ½) and to note that this limit is a strong equilibrium. A non-homogeneous Markov chain result is also used to establish the geometric rate of convergence. A proof of a Boltzmann H-theorem is also es

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38

Giroux, Gaston. "Asymptotic results for non-linear processes of the McKean tagged-molecule type." Journal of Applied Probability 23, no. 01 (1986): 42–51. http://dx.doi.org/10.1017/s0021900200106266.

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McKean's tagged-molecule process is a non-linear homogeneous two-state Markov chain in continuous time, constructed with the aid of a binary branching process. For each of a large class of branching processes we construct a similar process. The construction is carefully done and the weak homogeneity is deduced. A simple probability argument permits us to show convergence to the equidistribution (½, ½) and to note that this limit is a strong equilibrium. A non-homogeneous Markov chain result is also used to establish the geometric rate of convergence. A proof of a Boltzmann H-theorem is also es

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39

Pakes, Anthony G. "Convergence Rates and Limit Theorems for the Dual Markov Branching Process." Journal of Probability and Statistics 2017 (2017): 1–13. http://dx.doi.org/10.1155/2017/1410507.

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This paper studies aspects of the Siegmund dual of the Markov branching process. The principal results are optimal convergence rates of its transition function and limit theorems in the case that it is not positive recurrent. Additional discussion is given about specifications of the Markov branching process and its dual. The dualising Markov branching processes need not be regular or even conservative.

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40

Jacka, S. D., and G. O. Roberts. "Weak convergence of conditioned processes on a countable state space." Journal of Applied Probability 32, no. 4 (1995): 902–16. http://dx.doi.org/10.2307/3215203.

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We consider the problem of conditioning a continuous-time Markov chain (on a countably infinite state space) not to hit an absorbing barrier before time T; and the weak convergence of this conditional process as T → ∞. We prove a characterization of convergence in terms of the distribution of the process at some arbitrary positive time, t, introduce a decay parameter for the time to absorption, give an example where weak convergence fails, and give sufficient conditions for weak convergence in terms of the existence of a quasi-stationary limit, and a recurrence property of the original process

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41

Jacka, S. D., and G. O. Roberts. "Weak convergence of conditioned processes on a countable state space." Journal of Applied Probability 32, no. 04 (1995): 902–16. http://dx.doi.org/10.1017/s0021900200103377.

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We consider the problem of conditioning a continuous-time Markov chain (on a countably infinite state space) not to hit an absorbing barrier before time T; and the weak convergence of this conditional process as T → ∞. We prove a characterization of convergence in terms of the distribution of the process at some arbitrary positive time, t, introduce a decay parameter for the time to absorption, give an example where weak convergence fails, and give sufficient conditions for weak convergence in terms of the existence of a quasi-stationary limit, and a recurrence property of the original process

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42

Deng, C. S., R. L. Schilling, and Y. H. Song. "Subgeometric rates of convergence for Markov processes under subordination - Correction." Advances in Applied Probability 50, no. 3 (2018): 1005. http://dx.doi.org/10.1017/apr.2018.44.

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43

Gaudio, Julia, Saurabh Amin, and Patrick Jaillet. "Exponential convergence rates for stochastically ordered Markov processes under perturbation." Systems & Control Letters 133 (November 2019): 104515. http://dx.doi.org/10.1016/j.sysconle.2019.104515.

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44

Douc, Randal, Gersende Fort, and Arnaud Guillin. "Subgeometric rates of convergence of f-ergodic strong Markov processes." Stochastic Processes and their Applications 119, no. 3 (2009): 897–923. http://dx.doi.org/10.1016/j.spa.2008.03.007.

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45

Huang, Gang, Michel Mandjes, and Peter Spreij. "Weak convergence of Markov-modulated diffusion processes with rapid switching." Statistics & Probability Letters 86 (March 2014): 74–79. http://dx.doi.org/10.1016/j.spl.2013.12.013.

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46

Mao, Yong-Hua, Liping Xu, Ming Zhang, and Yu-Hui Zhang. "Convergence in total variation distance for (in)homogeneous Markov processes." Statistics & Probability Letters 137 (June 2018): 54–62. http://dx.doi.org/10.1016/j.spl.2018.01.011.

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47

Wang, Fengyu. "Coupling, convergence rates of Markov processes and weak Poincaré inequalities." Science in China Series A: Mathematics 45, no. 8 (2002): 975–83. http://dx.doi.org/10.1007/bf02879980.

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Alvarez-Mena, Jorge, and Onésimo Hernández-Lerma. "Convergence of the optimal values of constrained Markov control processes." Mathematical Methods of Operations Research 55, no. 3 (2002): 461–84. http://dx.doi.org/10.1007/s001860200209.

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Hart, Andrew G., and Richard L. Tweedie. "Convergence of Invariant Measures of Truncation Approximations to Markov Processes." Applied Mathematics 03, no. 12 (2012): 2205–15. http://dx.doi.org/10.4236/am.2012.312a301.

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Mufa, Chen. "ExponentialL 2-convergence andL 2-spectral gap for Markov processes." Acta Mathematica Sinica 7, no. 1 (1991): 19–37. http://dx.doi.org/10.1007/bf02582989.

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