In mathematics, a Markov chain, named after Andrey Markov, is a discrete-time stochastic process with the Markov property. Having the Markov property means that, given the present state, future states are independent of the past states. In other words, the present state description fully captures all the information that can influence the future evolution of the process. Thus, given the present, the future is conditionally independent of the past. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Andrey (Andrei) Andreyevich Markov (Russian: ) (June 14, 1856 N.S. – July 20, 1922) was a Russian mathematician. ... Discrete time is non-continuous time. ... In the mathematics of probability, a stochastic process is a random function. ... In probability theory, a stochastic process has the Markov property if the conditional probability distribution of future states of the process, given the present state, depends only upon the current state, i. ... In probability theory, two events A and B are conditionally independent given a third event C precisely if the occurrence or non-occurrence of A and B are independent events in their conditional probability distribution given C. Two random variables X and Y are conditionally independent given an event C...

At each time instant the system may change its state from the current state to another state, or remain in the same state, according to a certain probability distribution. The changes of state are called transitions, and the probabilities associated with various state-changes are termed transition probabilities. Also see sequential analysis. In statistics, sequential analysis refers to statistical analysis where the sample size is not fixed in advance. ...

Formal definition

A Markov chain is a sequence of random variables X₁, X₂, X₃, ... with the Markov property, namely that, given the present state, the future and past states are independent. Formally, In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In probability theory, a stochastic process has the Markov property if the conditional probability distribution of future states of the process, given the present state, depends only upon the current state, i. ...

The possible values of X_i form a countable set S called the state space of the chain. In mathematics, a countable set is a set with the same cardinality (i. ...

Markov chains are often described by a directed graph, where the edges are labeled by the probabilities of going from one state to the other states. This article just presents the basic definitions. ...

Variations

Continuous-time Markov processes have a continuous index. In probability theory, a continuous-time Markov process is a stochastic process { X(t) : t ≥ 0 } that satisfies the Markov property and takes values from a set called the state space. ...

Time-homogeneous Markov chains (or, Markov chains with time-homogeneous transition probabilities) are processes where

for all n.

A Markov chain of order m (or a Markov chain with memory m) where m is finite, is where

for all n. It is possible to construct a chain (Y_n) from (X_n) which has the 'classical' Markov property as follows: Let Y_n = (X_n, X_n−1, ..., X_n−m+1), the ordered m-tuple of X values. Then Y_n is a Markov chain with state space S^m and has the classical Markov property. In probability theory, a stochastic process has the Markov property if the conditional probability distribution of future states of the process, given the present state, depends only upon the current state, i. ... In probability theory, a stochastic process has the Markov property if the conditional probability distribution of future states of the process, given the present state, depends only upon the current state, i. ...

Example

A finite state machine can be used as a representation of a Markov chain. If the machine is in state y at time n, then the probability that it moves to state x at time n + 1 depends only on the current state. Fig. ...

A thorough development and many examples can be found in the on-line monograph Meyn & Tweedie 2005 ^[1] The appendix of Meyn 2007 ^[2], also available on-line, contains an abridged Meyn & Tweedie.

Properties of Markov chains

Define the probability of going from state i to state j in n time steps as

and the single-step transition as

The n-step transition satisfies the Chapman-Kolmogorov equation, that for any k such that 0 < k < n, In mathematics, specifically in probability theory, and yet more specifically in the theory of stochastic processes, the Chapman-Kolmogorov equation is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process. ...

The marginal distribution Pr (X_n = x) is the distribution over states at time n. The initial distribution is Pr (X₀ = x). The evolution of the process through one time step is described by In probability theory, given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y. For discrete random variables, the marginal probability mass function can...

The superscript (n) is intended to be an integer-valued label only; however, if the Markov chain is time-stationary, then this superscript can also be interpreted as a "raising to the power of", discussed further below.

Reducibility

A state j is said to be accessible from a different state i (written i → j) if, given that we are in state i, there is a non-zero probability that at some time in the future, we will be in state j. Formally, state j is accessible from state i if there exists an integer n≥0 such that

Allowing n to be zero means that every state is defined to be accessible from itself.

A state i is said to communicate with state j (written i ↔ j) if it is true that both i is accessible from j and that j is accessible from i. A set of states C is a communicating class if every pair of states in C communicates with each other, and no state in C communicates with any state not in C. (It can be shown that communication in this sense is an equivalence relation). A communicating class is closed if the probability of leaving the class is zero, namely that if i is in C but j is not, then j is not accessible from i. In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ...

Finally, a Markov chain is said to be irreducible if its state space is a communicating class; this means that, in an irreducible Markov chain, it is possible to get to any state from any state.

Periodicity

A state i has period k if any return to state i must occur in multiples of k time steps. For example, if it is only possible to return to state i in an even number of steps, then i is periodic with period 2. Formally, the period of a state is defined as

(where "gcd" is the greatest common divisor). Note that even though a state has period k, it may not be possible to reach the state in k steps. For example, suppose it is possible to return to the state in {6,8,10,12,...} time steps; then k would be 2, even though 2 does not appear in this list. In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...

If k = 1, then the state is said to be aperiodic; otherwise (k>1), the state is said to be periodic with period k.

It can be shown that every state in a communicating class must have the same period.

A finite state irreducible Markov chain is said to be ergodic if its states are aperiodic.

Recurrence

A state i is said to be transient if, given that we start in state i, there is a non-zero probability that we will never return back to i. Formally, let the random variable T_i be the first return time to state i (the "hitting time"): In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...

Then, state i is transient iff: IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

If a state i is not transient (it has finite hitting time with probability 1), then it is said to be recurrent or persistent. Although the hitting time is finite, it need not have a finite expectation. Let M_i be the expected return time, In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

Then, state i is positive recurrent if M_i is finite; otherwise, state i is null recurrent (the terms non-null persistent and null persistent are also used, respectively).

It can be shown that a state is recurrent if and only if ↔ ⇔ ≡ logical symbols representing iff. ...

A state i is called absorbing if it is impossible to leave this state. Therefore, the state i is absorbing if and only if

p_ii = 1 and p_ij = 0 for

Ergodicity

A state i is said to be ergodic if it is aperiodic and positive recurrent. If all states in a Markov chain are ergodic, then the chain is said to be ergodic. In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ...

Steady-state analysis and limiting distributions

If the Markov chain is a time-homogeneous Markov chain, so that the process is described by a single, time-independent matrix p_ij, then the vector π is a stationary distribution (also called an equilibrium distribution or invariant measure) if its entries π_j sum to 1 and satisfy In mathematics, an invariant measure is a measure that is preserved by some function. ...

An irreducible chain has a stationary distribution if and only if all of its states are positive-recurrent. In that case, π is unique and is related to the expected return time:

Further, if the chain is both irreducible and aperiodic, then for any i and j,

Note that there is no assumption on the starting distribution; the chain converges to the stationary distribution regardless of where it begins.

If a chain is not irreducible, its stationary distributions will not be unique (consider any closed communicating class in the chain; each one will have its own unique stationary distribution. Any of these will extend to a stationary distribution for the overall chain, where the probability outside the class is set to zero). However, if a state j is aperiodic, then

and for any other state i, let f_ij be the probability that the chain ever visits state j if it starts at i,

Markov chains with a finite state space

If the state space is finite, the transition probability distribution can be represented by a matrix, called the transition matrix, with the (i, j)'th element of P equal to In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

P is a stochastic matrix. Further, when the Markov chain is a time-homogeneous Markov chain, so that the transition matrix P is independent of the label n, then the k-step transition probability can be computed as the k'th power of the transition matrix, P^k. In mathematics, a stochastic matrix, probability matrix, or transition matrix is used to describe the transitions of a Markov chain. ...

The stationary distribution π is a (row) vector which satisfies the equation

In other words, the stationary distribution π is a normalized left eigenvector of the transition matrix associated with the eigenvalue 1. In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

Alternatively, π can be viewed as a fixed point of the linear (hence continuous) transformation on the unit simplex associated to the matrix P. As any continuous transformation in the unit simplex has a fixed point, a stationary distribution always exists, but is not guaranteed to be unique, in general. However, if the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution π. In addition, P^k converges to a rank-one matrix in which each row is the stationary distribution π, that is, A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ...

where 1 is the column vector with all entries equal to 1. This is stated by the Perron-Frobenius theorem. This means that as time goes by, the Markov chain forgets where it began (its initial distribution) and converges to its stationary distribution. In mathematics, the Perron-Frobenius theorem is a theorem in matrix theory about the eigenvalues and eigenvectors of a real positive n×n matrix: Let A = (aij) be a real n×n matrix with positive entries . ...

Reversible Markov chain

The idea of a reversible Markov chain comes from the ability to "invert" a conditional probability using Bayes' Rule: Bayes theorem is a result in probability theory, which gives the conditional probability distribution of a random variable A given B in terms of the conditional probability distribution of variable B given A and the marginal probability distribution of A alone. ...

It now appears that time has been reversed. Thus, a Markov chain is said to be reversible if there is a π such that

This condition is also known as the detailed balance condition. In mathematics, and in statistical mechanics in physics, a Markov process is said to show detailed balance if the transition rates between each pair of states i and j in the state space obey where P is the Markov transition matrix (transition probability), ie Pij = P( Xt =j | Xt−1...

Summing over i gives

so for reversible Markov chains, π is always a stationary distribution.

Bernoulli scheme

A Bernoulli scheme is a special case of a Markov chain where the transition probability matrix has identical rows, which means that the next state is even independent of the current state (in addition to being independent of the past states). A Bernoulli scheme with only two possible states is known as a Bernoulli process. In mathematics, the Bernoulli scheme is a generalization of the Bernoulli process to more than two possible outcomes. ... In probability and statistics, a Bernoulli process is a discrete-time stochastic process consisting of a sequence of independent random variables taking values over two symbols. ...

Markov chains with general state space

Many results for Markov chains with finite state space can be generalized to chains with uncountable state space through Harris chains. The main idea is to see if there is a point in the state space that the chain hits with probability one. Generally, it is not true for continuous state space, however, we can define sets A and B along with a positive number ε and a probability measure ρ, such that

1. If , then for all z.
2. If and , then.

Then we could collapse the sets into an auxiliary point α, and a recurrent Harris chain can be modified to contain α. Lastly, the collection of Harris chains is a comfortable level of generality, which is broad enough to contain a large number of interesting examples, yet restrictive enough to allow for a rich theory.

Applications

Physics

Markovian systems appear extensively in physics, particularly statistical mechanics, whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...

Testing

Several theorists have proposed the idea of the Markov chain statistical test, a method of conjoining Markov chains to form a 'Markov blanket', arranging these chains in several recursive layers ('wafering') and producing more efficient test sets — samples — as a replacement for exhaustive testing. MCSTs also have uses in temporal state-based networks; Chilukuri et al.'s paper entitled "Temporal Uncertainty Reasoning Networks for Evidence Fusion with Applications to Object Detection and Tracking" (ScienceDirect) gives an excellent background and case study for applying MCSTs to a wider range of applications.

Queueing theory

Markov chains can also be used to model various processes in queueing theory and statistics.^[2]. Claude Shannon's famous 1948 paper A mathematical theory of communication, which at a single step created the field of information theory, opens by introducing the concept of entropy through Markov modeling of the English language. Such idealised models can capture many of the statistical regularities of systems. Even without describing the full structure of the system perfectly, such signal models can make possible very effective data compression through entropy coding techniques such as arithmetic coding. They also allow effective state estimation and pattern recognition. The world's mobile telephone systems depend on the Viterbi algorithm for error-correction, while hidden Markov models are extensively used in speech recognition and also in bioinformatics, for instance for coding region/gene prediction. Markov chains also play an important role in reinforcement learning. Queueing theory (also commonly spelled queuing theory) is the mathematical study of waiting lines (or queues). ... This article is about the field of statistics. ... Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001), an American electrical engineer and mathematician, has been called the father of information theory,[1] and was the founder of practical digital circuit design theory. ... The article entitled A Mathematical Theory of Communication, published in 1948 by mathematician Claude E. Shannon, was one of the founding works of the field of information theory. ... Not to be confused with information technology, information science, or informatics. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... Source coding redirects here. ... An entropy encoding is a coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Pattern recognition is a field within the area of machine learning. ... The Viterbi algorithm, named after its developer Andrew Viterbi, is a dynamic programming algorithm for finding the most likely sequence of hidden states – known as the Viterbi path – that result in a sequence of observed events, especially in the context of hidden Markov models. ... A hidden Markov model (HMM) is a statistical model where the system being modelled is assumed to be a Markov process with unknown parameters, and the challenge is to determine the hidden parameters, from the observable parameters, based on this assumption. ... Speech recognition (in many contexts also known as automatic speech recognition, computer speech recognition or erroneously as voice recognition) is the process of converting a speech signal to a sequence of words in the form of digital data, by means of an algorithm implemented as a computer program. ... Map of the human X chromosome (from the NCBI website). ... Reinforcement learning refers to a class of problems in machine learning which postulate an agent exploring an environment in which the agent perceives its current state and takes actions. ...

Internet applications

The PageRank of a webpage as used by Google is defined by a Markov chain. It is the probability to be at page i in the stationary distribution on the following Markov chain on all (known) webpages. If N is the number of known webpages, and a page i has k_i links then it has transition probability for all pages that are linked to and for all pages that are not linked to. The parameter q is taken to be about 0.15. Mathematical PageRanks (out of 100) for a simple network (PageRanks reported by google are rescaled logarithmically). ... This article is about the corporation. ...

Markov models have also been used to analyze web navigation behavior of users. A user's web link transition on a particular website can be modeled using first- or second-order Markov models and can be used to make predictions regarding future navigation and to personalize the web page for an individual user.

Statistical

Markov chain methods have also become very important for generating sequences of random numbers to accurately reflect very complicated desired probability distributions, via a process called Markov chain Monte Carlo (MCMC). In recent years this has revolutionised the practicability of Bayesian inference methods, allowing a wide range of posterior distributions to be simulated and their parameters found numerically. Markov chain Monte Carlo (MCMC) methods (which include random walk Monte Carlo methods) are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its stationary distribution. ... Bayesian inference is statistical inference in which evidence or observations are used to update or to newly infer the probability that a hypothesis may be true. ... In Bayesian probability theory, the posterior probability is the conditional probability of some event or proposition, taking empirical data into account. ...

Economics

Dynamic macroeconomics heavily uses Markov chain.

Mathematical biology

Markov chains also have many applications in biological modelling, particularly population processes, which are useful in modelling processes that are (at least) analogous to biological populations. The Leslie matrix is one such example, though some of its entries are not probabilities (they may be greater than 1). Another important example is the modeling of cell shape in dividing sheets of epithelial cells]. The distribution of shapes -- predominantly hexagonal -- was a long standing mystery until it was explained by a simple Markov Model, where a cell's state is its number of sides. Empirical evidence from frogs, fruit flies, and hydra further suggests that the stationary distribution of cell shape is exhibited by almost all multicellular animals.[1] In applied probability, a population process is a Markov chain in which the state of the chain is analogous to the number of individuals in a population (0, 1, 2, etc. ... The Leslie Matrix is a discrete and age-structured model of population growth very popular in population ecology. ... In zootomy, epithelium is a tissue composed of a layer of cells. ...

Gambling

Markov chains can be used to model many games of chance. The children's games Snakes and Ladders and "Hi Ho! Cherry-O", for example, are represented exactly by Markov chains. At each turn, the player starts in a given state (on a given square) and from there has fixed odds of moving to certain other states (squares). Snakes and ladders, or Chutes and ladders, is a classic childrens board game. ... Hi Ho! Cherry-O is a board game made for preschool children by Hasbro of Pawtucket, Rhode Island, under its subsidiary, Milton Bradley. ...

Music

Markov chains are employed in algorithmic music composition, particularly in software programs such as CSound or Max. In a first-order chain, the states of the system become note or pitch values, and a probability vector for each note is constructed, completing a transition probability matrix (see below). An algorithm is constructed to produce and output note values based on the transition matrix weightings, which could be MIDI note values, frequency (Hz), or any other desirable metric. It has been suggested that Generative music be merged into this article or section. ... Computer software (or simply software) refers to one or more computer programs and data held in the storage of a computer for some purpose. ... Csound is a computer programming language for dealing with sound, also known as a sound compiler or a music programming language. ... A Max/MSP patch written and used by Autechre Max is a graphical development environment for music and multimedia developed and maintained by San Francisco-based software company Cycling 74. ... In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one. ... Musical Instrument Digital Interface, or MIDI, is a system designed to transmit information between electronic musical instruments. ... Hz or hz may mean: Herero language (ISO 639 alpha-2, hz) Hertz, unit of frequency This is a disambiguation page — a list of articles associated with the same title. ...

A second-order Markov chain can be introduced by considering the current state and also the previous state, as indicated in the second table. Higher, nth-order chains tend to "group" particular notes together, while 'breaking off' into other patterns and sequences occasionally. These higher-order chains tend to generate results with a sense of phrasal structure, rather than the 'aimless wandering' produced by a first-order system^[3]. In music a phrase (Greek φράση, sentence, expression, see also strophe) is a section of music that is relatively self contained and coherent over a medium time scale. ...

Baseball

Markov chains models have been used in advanced baseball analysis since 1960, although their use is still rare. Each half-inning of a baseball game fits the Markov chain state when the number of runners and outs are considered. For each half-inning there are 24 possible run-out combinations. Markov chain models can be used to evaluate runs created for both individual players as well as a team. ^[4].

Markov parody generators

Markov processes can also be used to generate superficially "real-looking" text given a sample document: they are used in a variety of recreational "parody generator" software (see dissociated press, Jeff Harrison, Mark V Shaney, ^{[5] [6]} ). Dissociated Press Play on ‘Associated Press’; perhaps inspired by a reference in the 1950 Bugs Bunny cartoon Whats Up, Doc? An algorithm for transforming any text into potentially humorous garbage even more efficiently than by passing it through a marketroid. ... Jeff Harrison is an American poet whose poems Postmortem Series and Accuracy are apparently randomly generated - a form of aleatoric poetry. ... Mark V Shaney is a fake Usenet user whose postings were generated by using Markov chain techniques. ...

History

Andrey Markov produced the first results (1906) for these processes, purely theoretically. A generalization to countably infinite state spaces was given by Kolmogorov (1936). Markov chains are related to Brownian motion and the ergodic hypothesis, two topics in physics which were important in the early years of the twentieth century, but Markov appears to have pursued this out of a mathematical motivation, namely the extension of the law of large numbers to dependent events. In 1913, he applied his findings for the first time, namely, to the first 20,000 letters of Pushkin's "Eugene Onegin". Andrey (Andrei) Andreyevich Markov (Russian: ) (June 14, 1856 N.S. – July 20, 1922) was a Russian mathematician. ... Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Russian mathematician who made major advances in the fields of probability theory and topology. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i. ... // The law of large numbers (LLN) is any of several theorems in probability. ...

See also

State transitions in a hidden Markov model (example) x — hidden states y — observable outputs a — transition probabilities b — output probabilities A hidden Markov model (HMM) is a statistical model in which the system being modeled is assumed to be a Markov process with unknown parameters, and the challenge is to... This page contains examples of Markov chains in action. ... It has been suggested that this article or section be merged with Markov property. ... Markov chain Monte Carlo (MCMC) methods (which include random walk Monte Carlo methods) are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its stationary distribution. ... A semi-Markov process is one that, when it enters state i, spends a random time having distribution and mean in that state before making a transition. ... Variable-order Markov (VOM) models are an important class of models that extend the well known Markov chain models. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. ... Mark V Shaney is a fake Usenet user whose postings were generated by using Markov chain techniques. ... A phase-type distribution is a probability distribution that results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. ... In mathematical probability, a fundamental result about Markov chains is that a finite state irreducible aperiodic chain has a unique stationary distribution π and, regardless of the initial state, the time-t distribution of the chain converges to π as t tends to infinity. ... In mathematics, the quantum Markov chain is a reformulation of the ideas of a classical Markov chain, replacing the classical definitions of probability with quantum probability. ... A Markov network, or Markov random field, is a model of the (full) joint probability distribution of a set of random variables. ... Belief propagation is an iterative algorithm for computing marginals of functions on a graphical model most commonly used in artificial intelligence and information theory. ... A factor graph is an -bipartite graph where is a set of variables and is a set of factors. ... In dynamical systems, stochastic processes, and time series analysis, recurrence period density entropy (RPDE) is a method for determining the periodicity, or repetitiveness of a signal. ...

References

• A.A. Markov. "Rasprostranenie zakona bol'shih chisel na velichiny, zavisyaschie drug ot druga". Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 2-ya seriya, tom 15, pp 135-156, 1906.

• A.A. Markov. "Extension of the limit theorems of probability theory to a sum of variables connected in a chain". reprinted in Appendix B of: R. Howard. Dynamic Probabilistic Systems, volume 1: Markov Chains. John Wiley and Sons, 1971.

• Classical Text in Translation: A. A. Markov, An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains, trans. David Link. Science in Context 19.4 (2006): 591-600. Online: http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=637500

• Leo Breiman. Probability. Original edition published by Addison-Wesley, 1968; reprinted by Society for Industrial and Applied Mathematics, 1992. ISBN 0-89871-296-3. (See Chapter 7.)

• J.L. Doob. Stochastic Processes. New York: John Wiley and Sons, 1953. ISBN 0-471-52369-0.

• S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. London: Springer-Verlag, 1993. ISBN 0-387-19832-6. online: http://decision.csl.uiuc.edu/~meyn/pages/book.html . Second edition to appear, Cambridge University Press, 2008.

• S. P. Meyn. Control Techniques for Complex Networks. Cambridge University Press, 2007. ISBN-13: 9780521884419. Appendix contains abridged Meyn & Tweedie. online: http://decision.csl.uiuc.edu/~meyn/pages/CTCN/CTCN.html

• Booth, Taylor L. (1967). Sequential Machines and Automata Theory, 1st, New York: John Wiley and Sons, Inc.. Library of Congress Card Catalog Number 67-25924.  Extensive, wide-ranging book meant for specialists, written for both theoretical computer scientists as well as electrical engineers. With detailed explanations of state minimization techniques, FSMs, Turing machines, Markov processes, and undecidability. Excellent treatment of Markov processes pp.449ff. Discusses Z-transforms, D transforms in their context.

• Kemeny, John G.; Hazleton Mirkil, J. Laurie Snell, Gerald L. Thompson (1959). Finite Mathematical Structures, 1st, Englewood Cliffs, N.J.: Prentice-Hall, Inc.. Library of Congress Card Catalog Number 59-12841.  Classical text. cf Chapter 6 Finite Markov Chains pp.384ff.

Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 330th day of the year (331st in leap years) in the Gregorian calendar. ...

External links

• (pdf) Markov Chains chapter in American Mathematical Society's introductory probability book
• Generates random parodies in the style of another body of text using a Markov chain algorithm
• A generator that uses Markov Chains to create random words
• Markov Chains
• Class structure on PlanetMath
• Chapter 5: Markov Chain Models
• Generating Text (About generating random text using a Markov chain.)
• The World's Largest Matrix Computation (Google's PageRank as the stationary distribution of a random walk through the Web.)
• Dissociated Press in Emacs approximates a Markov process
• Markov Chain Example
• Markov Chains for Search Engines
• Steganography proof-of-concept using Markov Chains.
• n^th order Markov Chain implementation in Ruby
• Baseball Run Modeler using Markov Chains
• Theory of Markov chains in baseball
• Sequential analysis software for generating visual representations of probability models