Markov chain

A Markov chain $(𝒦, 𝑆)$ is a random sequence of events where the probability of each event depends only on the previous event occurs. Thus such a process may be characterized by a state space $𝑆$ and a transition matrix $𝒦 : 𝑆 \times 𝑆 \to [0, 1]$ , such that for each $𝑠 \in 𝑆$ we have

\sum 𝑠' \in 𝑆 𝒦 (𝑠, 𝑠') = 1

A stationary distribution is a distribution of initial states which remains invariant upon transition, i.e. an eigenvector of eigenvalue 1 with entries summing to 1.

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