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In probability theory, the matrix analytic method is a technique to compute the stationary probability distribution of a Markov chain which has a repeating structure (after some point) and a state space which grows unboundedly in no more than one dimension.
For a continuous time Markov chain (CTMC) with transition rate matrix, if can be found such that for every pair of states and = holds, then by summing over , the global balance equations are satisfied and is the stationary distribution of the process. [5]
Consider a finite state irreducible aperiodic Markov chain with state space and (unique) stationary distribution (is a probability vector). Suppose that we come up with a probability distribution on the set of maps : with the property that for every fixed , its image () is distributed according to the transition probability of from state .
D. G. Champernowne built a Markov chain model of the distribution of income in 1953. [93] Herbert A. Simon and co-author Charles Bonini used a Markov chain model to derive a stationary Yule distribution of firm sizes. [94] Louis Bachelier was the first to observe that stock prices followed a random walk. [95]
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]
A Markov chain is said to be irreducible when every state can reach every other state through some sequence of transitions, and aperiodic if, for every state, the possible numbers of steps in sequences that start and end in that state have greatest common divisor one. An irreducible and aperiodic Markov chain necessarily has a stationary ...
We say is Markov with initial distribution and rate matrix to mean: the trajectories of are almost surely right continuous, let be a modification of to have (everywhere) right-continuous trajectories, (()) = + almost surely (note to experts: this condition says is non-explosive), the state sequence (()) is a discrete-time Markov chain with ...
A Markov process is called a reversible Markov process or reversible Markov chain if there exists a positive stationary distribution π that satisfies the detailed balance equations [13] =, where P ij is the Markov transition probability from state i to state j, i.e. P ij = P(X t = j | X t − 1 = i), and π i and π j are the equilibrium probabilities of being in states i and j, respectively ...