By R. L. Goodstein

ISBN-10: 0236176692

ISBN-13: 9780236176694

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**Extra info for Development of Mathematical Logic**

**Example text**

For all i, j ∈ S, we have: i −→ j and i positive recurrent =⇒ j positive recurrent. – If i = j the result is trivial. Let i and j be two states such that i = j and let us assume that i −→ j and that i is positive recurrent. 13, we know that j is recurrent and that fi,j = fj,i = 1. 1, we have j −→ i. Since i = j, there exist two integers ≥ 1 and m ≥ 1 such that: (P )i,j > 0 and (P m )j,i > 0. It then follows that, for all k ≥ 0, we have: (P m+k+ )j,j ≥ (P m )j,i (P k )i,i (P )i,j . Summing over k from 0 to n − 1 with n ≥ 1 and dividing by n, we obtain: 1 n n−1 (P m+k+ )j,j ≥ (P m )j,i (P )i,j k=0 1 n n−1 (P k )i,i k=0 and after a change of variable in the left hand side: 1 n m+ +n−1 k=0 (P k )j,j − 1 n m+ −1 (P k )j,j ≥ (P m )j,i (P )i,j k=0 1 n n−1 (P k )i,i .

An invariant probability on S is an invariant measure v = (vj , j ∈ S) such that v = 1. The invariant probability v is said to be positive if vj > 0, for all j ∈ S. – Let X be an irreducible Markov chain. X is positive recurrent if and only if it has an invariant probability. In this case, the invariant probability is unique and positive, we denote it by π = (πj , j ∈ S) and it is given by: πj = 1/mj . – Let X be an irreducible Markov chain. 17, it has, up to a multiplicative constant, a unique positive invariant measure.

We have shown that the process W = {Wn , n ∈ } deﬁned by Wn = (Xn , Yn ) is a Markov chain on S × S and that the aperiodicity of X results in the irreducibility of W . 1, we have, for all j ∈ S, lim n−→∞ {Wn = (j, j)} = 0. Moreover, taking μ = α, we have, for all j ∈ S, 2 [ {Xn = j}] = {Xn = j, Yn = j} = hence: lim n−→∞ {Xn = j} = 0, {Wn = (j, j)}, Discrete-Time Markov Chains 39 which completes the proof in this case. 22, we have, for every initial distribution α of X and μ of Y , lim | {Xn = j} − n−→∞ {Yn = j}| = 0, and, for all k ≥ 0, lim | {Xn−k = j} − n−→∞ {Yn−k = j}| = 0.

### Development of Mathematical Logic by R. L. Goodstein

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