finite state markov chain l finite stochastic matrix then there exists an invariant distribution

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Finite-state Markov chain: Let L be finite. A stochastic matrix: Then there exists an invariant distribution by Brouwer's fixed-point theorem, which is a compact convex subset of Rd. Let f: R^d -> R^d be a continuous mapping. Then there is a point Xo â‚¬ S such that f(xo) = xo (fixed point). For example, let f: [0,1] -> [0,1] be continuous. Then there is a point x â‚¬ [0,1] such that f(x) = x. Without loss of generality, assume that L = {1, d}. Consider S = {A = (x1, x2, ..., xd) â‚¬ Rd : xi >= 0, Î£xi = 1}. In fact, S is the space of distributions on Z. See the figure in case d = 2. S is compact and convex. Given P, the map f: S -> S defined by f(A) = HAP is continuous. Therefore, there is a fixed point T such that f(T) = T.

          Finite-state Markov chain:
Let L be finite. A stochastic matrix: Then there exists an invariant distribution by Brouwer's fixed-point theorem, which is a compact convex subset of Rd. Let f: R^d -> R^d be a continuous mapping. Then there is a point Xo â‚¬ S such that f(xo) = xo (fixed point).

For example, let f: [0,1] -> [0,1] be continuous. Then there is a point x â‚¬ [0,1] such that f(x) = x. Without loss of generality, assume that L = {1, d}. Consider S = {A = (x1, x2, ..., xd) â‚¬ Rd : xi >= 0, Î£xi = 1}. In fact, S is the space of distributions on Z. See the figure in case d = 2. S is compact and convex. Given P, the map f: S -> S defined by f(A) = HAP is continuous. Therefore, there is a fixed point T such that f(T) = T.

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