Proof to check this claim we note that if x 1n then x

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Unformatted text preview: fit is 11.54 2.43 = 9.11 dollars per day. At this point we have computed 2· policy profit per day 0,3 $9.11 1,3 $9.62 2,3 $9.40 so the 1,3 inventory policy is optimal. 1.6 1.6.1 Special Examples Doubly stochastic chains Definition 1.2. A transition matrixP is said to be doubly stochastic if its p COLUMNS sum to 1, or in symbols x p(x, y ) = 1. The adjective “doubly” refers to the fact that by its definition a transition probP ability matrix has ROWS that sum to 1, i.e., y p(x, y ) = 1. The stationary distribution is easy to guess in this case: Theorem 1.24. If p is a doubly stochastic transition probability for a Markov chain with N states, then the uniform distribution, ⇡ (x) = 1/N for all x, is a stationary distribution. Proof. To check this claim we note that if ⇡ (x) = 1/N then X 1X 1 = ⇡ (y ) ⇡ (x)p(x, y ) = p(x, y ) = Nx N x Looking at the second equality we see that conversely, if ⇡ (x) = 1/N then p is doubly stochastic. Example 1.25. Symmetric reflecting random walk on the line. The state space is {0, 1,...
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