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...tion source]] whose underlying dynamics are given by a stationary finite [[Markov chain]]. A Markov information source is then a (stationary) Markov chain <math>M</math>, together with a [[Function (mathematics)|function]] ...

{{ about|the theorem in Markov probability theory|the theorem in electrical engineering|Foster's reactance ...with [[countable]] state spaces. It uses the fact that positive recurrent Markov chains exhibit a notion of "[[Lyapunov stability]]" in terms of returning t ...

...cading dependence.<ref>https://ak2316.user.srcf.net/files/ib-markov-chains/markov-chains.pdf</ref> # <math>\theta_k^1</math> is a [[Markov chain]] with transition probability matrix <math>\Lambda^1</math> ...

...a measurable state space''' is a [[Markov chain|discrete-time-homogeneous Markov chain]] with a [[measure (mathematics)|measurable space]] as state space. ...tic Stability''. 2nd edition, 2009.</ref><ref name="revuz">Daniel Revuz: ''Markov Chains''. 2nd edition, 1984.</ref><ref>Rick Durrett: ''Probability: Theory ...

{{about|Kolmogorov's criterion in the study of Markov chains|Kolmogorov's criterion in the study of norms on topological vector s ...ssary and sufficient condition for a [[Markov chain]] or [[continuous-time Markov chain]] to be stochastically identical to its time-reversed version. ...

...method for reducing the size of the state space of some [[continuous-time Markov chain]]s, first published by [[John George Kemeny|Kemeny]] and Snell.<ref> | title = Finite Markov Chains ...

{{Short description|Matrix describing continuous-time Markov chains}} ...of numbers describing the instantaneous rate at which a [[continuous-time Markov chain]] transitions between states. ...

...d.{{r|williams}} It was first stated by {{harvtxt|Hill|1966}}, for certain Markov chains arising in [[thermodynamics]],{{r|williams|hill}} and proved in full ...t state have [[greatest common divisor]] one. An irreducible and aperiodic Markov chain necessarily has a stationary distribution, a probability distribution ...

=== In terms of Markov kernels=== ...compose]]. Given a [[measurable space]] <math>(X,\mathcal{A})</math> and a Markov kernel <math>k:(X,\mathcal{A})\to(X,\mathcal{A})</math>, the ''two-step tra ...

.... | last = Asmussen| doi = 10.1007/0-387-21525-5_2 | chapter = Markov Jump Processes | title = Applied Probability and Queues | series = Stochastic Modelling an For a continuous time Markov chain with state space ''S'' and [[transition-rate matrix]] ''Q'' (with ele ...

...Discrete-time stochastic process|discrete-time]] [[Markov chain#Variations|Markov chain of order ''m'']] and the transition probability to a state at the nex ...iables are fixed depends on the values of ''m'' previous variables only ([[Markov chain]] of order ''m''), and the influence of previous variables on a gener ...

...e mathematical theory of [[stochastic processes|random processes]], the '''Markov chain central limit theorem''' has a conclusion somewhat similar in form to ...om element]]s of some set is a [[Markov chain]] that has a [[Discrete-time Markov chain#Stationary distributions|stationary probability distribution]]; and ...

{{Short description|Type of stochastic Markov process}} .... The theory of Harris chains and Harris recurrence is useful for treating Markov chains on general (possibly uncountably infinite) state spaces. ...

...le=''Almost None of the Theory of Stochastic Processes: A Course on Random Processes, for Students of Measure-Theoretic Probability, with a View to Applications be the [[Markov kernel]] derived from <math> (\Omega^{i-1}, \mathcal A^{i-1}) </math> and < ...

...n Technique as a Modeling Tool and Solution Procedure for Transient Markov Processes | journal = Operations Research | volume = 32 | issue = 2 | pages = 343–361 ...h>, which is defined by<ref name="stewart">{{cite book |title=Probability, Markov chains, queues, and simulation: the mathematical basis of performance model ...

'''The decentralized partially observable Markov decision process (Dec-POMDP)'''<ref>{{Cite journal | title=The Complexity of Decentralized Control of Markov Decision Processes ...

{{Short description|Equations characterizing continuous-time Markov processes}} ...]es. In particular, they describe how the probability of a continuous-time Markov process in a certain state changes over time. ...

...f>{{Cite book | last1 = Reiss | first1 = R. D. | title = A Course on Point Processes | doi = 10.1007/978-1-4613-9308-5 | series = Springer Series in Statistics ...al A)</math> and <math>(Y,\mathcal B)</math> be [[measurable space]]s. A ''Markov kernel'' with source <math>(X,\mathcal A)</math> and target <math>(Y,\mathc ...

...lization of the [[Cauchy–Schwarz inequality]] to integrals of [[stochastic processes]]. ...''N'' be continuous [[local martingale]]s and ''H'', ''K'' [[measurable]] processes. Then ...

...s for Computing Stationary Distributions of Nearly Completely Decomposable Markov Chains | journal = [[SIAM Journal on Algebraic and Discrete Methods]] | vol A [[Markov chain]] with [[Stochastic matrix|transition matrix]] ...