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A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set (often representing time), but the precise definition of a Markov chain varies. [6]
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.
The stochastic matrix was developed alongside the Markov chain by Andrey Markov, a Russian mathematician and professor at St. Petersburg University who first published on the topic in 1906. [3] His initial intended uses were for linguistic analysis and other mathematical subjects like card shuffling , but both Markov chains and matrices rapidly ...
In the mathematical study of stochastic processes, a Harris chain is a Markov chain where the chain returns to a particular part of the state space an unbounded number of times. [1] Harris chains are regenerative processes and are named after Theodore Harris. The theory of Harris chains and Harris recurrence is useful for treating Markov chains ...
A basic property about an absorbing Markov chain is the expected number of visits to a transient state j starting from a transient state i (before being absorbed). This can be established to be given by the (i, j) entry of so-called fundamental matrix N, obtained by summing Q k for all k (from 0 to ∞).
Kemeny wrote, (for i the starting state of the Markov chain) “A prize is offered for the first person to give an intuitively plausible reason for the above sum to be independent of i.” [2] Grinstead and Snell offer an explanation by Peter Doyle as an exercise, with solution “he got it!” [8] [9]
Definition. A distribution on {,,,...} is a discrete phase-type distribution if it is the distribution of the first passage time to the absorbing state of a terminating Markov chain with finitely many states.
The Markov-modulated Poisson process or MMPP where m Poisson processes are switched between by an underlying continuous-time Markov chain. [8] If each of the m Poisson processes has rate λ i and the modulating continuous-time Markov has m × m transition rate matrix R, then the MAP representation is