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  2. Stochastic matrix - Wikipedia

    en.wikipedia.org/wiki/Stochastic_matrix

    The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. There are several different definitions and ...

  3. Markov chain - Wikipedia

    en.wikipedia.org/wiki/Markov_chain

    If a Markov chain has a stationary distribution, then it can be converted to a measure-preserving dynamical system: Let the probability space be =, where is the set of all states for the Markov chain. Let the sigma-algebra on the probability space be generated by the cylinder sets.

  4. Markov Chains and Mixing Times - Wikipedia

    en.wikipedia.org/wiki/Markov_Chains_and_Mixing_Times

    A family of Markov chains is said to be rapidly mixing if the mixing time is a polynomial function of some size parameter of the Markov chain, and slowly mixing otherwise. This book is about finite Markov chains, their stationary distributions and mixing times, and methods for determining whether Markov chains are rapidly or slowly mixing. [1] [4]

  5. Perron–Frobenius theorem - Wikipedia

    en.wikipedia.org/wiki/Perron–Frobenius_theorem

    The theorem has a natural interpretation in the theory of finite Markov chains (where it is the matrix-theoretic equivalent of the convergence of an irreducible finite Markov chain to its stationary distribution, formulated in terms of the transition matrix of the chain; see, for example, the article on the subshift of finite type).

  6. Discrete-time Markov chain - Wikipedia

    en.wikipedia.org/wiki/Discrete-time_Markov_chain

    A Markov chain with two states, A and E. In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.

  7. Markov model - Wikipedia

    en.wikipedia.org/wiki/Markov_model

    The simplest Markov model is the Markov chain.It models the state of a system with a random variable that changes through time. In this context, the Markov property indicates that the distribution for this variable depends only on the distribution of a previous state.

  8. Subshift of finite type - Wikipedia

    en.wikipedia.org/wiki/Subshift_of_finite_type

    A common object of study is the Markov measure, which is an extension of a Markov chain to the topology of the shift. A Markov chain is a pair ( P , π) consisting of the transition matrix , an n × n matrix P = ( p ij ) for which all p ij ≥ 0 and

  9. Transition matrix - Wikipedia

    en.wikipedia.org/wiki/Transition_matrix

    Stochastic matrix, a square matrix used to describe the transitions of a Markov chain. State-transition matrix , a matrix whose product with the state vector x {\displaystyle x} at an initial time t 0 {\displaystyle t_{0}} gives x {\displaystyle x} at a later time t {\displaystyle t} .