Markov Chain (Discrete Time)

1. Definition

Let {X_n, n =0,1,cdots} be a stochastic process, taking on a finite or countable number of values.

X_n is a DTMC if it has the Markov property: Given the present, the future is independent of the past
P{X_{n+1} = j| X_n = j, X_{n-1} = i_{n-1} ,cdots, X_1 = i_1, X_0 = i_0} = Pr{X_{n+1} = j| X_n = i}

We define Pr{X_{n+1} = j| X_n = i} = P_{ij}, since X_n has the stationary transition probabilities, this probability is not depend on n.

Transition probabilities satisfy sum_j p_{ij} = 1

2. n Step transition Probabilities

P^{n+m}_{ij} = sum_k p^m_{kj} p^n_{ik}


3. Example: Coin Flips

4. Limiting Probabilities

Theorem: For an irreducible, ergodic Markov Chain, pi_j = lim_{n to infty} P^n_{ij} exists and is independent of the starting state i. Then pi_j is the unique solution of pi_j = sum_i pi_i P_{ij} and sum_j pi_j = 1.

Two interpretation for pi_i

  • The probability of being in state i a long time into the future (large n)
  • The long-run fraction of time in state i
  • If Markov Chain is irreducible and ergodic, then interpretation 1 and 2 are equivalent
  • Otherwise, pi_i is still the solution to pi = pi P, but only interpretation 2 is valid.

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