For the Markov chain given below what is the best way to find the probability P{$x_n = x_{n-1}$} and P{$x_n \neq x_{n-1}$}
The transition matrix of the chain is \begin{array}{} 1/3 & 2/3 & 0 \\ 0 & 2/3 & 1/3 \\ 0 & 2/3 & 1/3 \end{array}
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1Hello. Do you know what the distribution of $x_{n-1}$ is? – Sam Livingstone Apr 15 '14 at 07:25
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i am sorry , i forgot to add that the initial distributions are given: P(x_0 = 0)= 1/2, P(x_0 =1)= 1/2 and P(x_0 = 2)=0. – user143234 Apr 15 '14 at 07:27
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Actually we don't care... – Did Apr 15 '14 at 07:29
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Why are you revising your question, omitting conditions that where previously specified? – Did Apr 15 '14 at 07:38
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For every state $x$, $P(x_1=x\mid x_0=x)=\frac13$, hence $P(x_n=x_{n-1})=\frac13$ and $P(x_n\ne x_{n-1})=\frac23$, for every initial distribution and every $n$.
Did
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thanks, but what if the probabilities are not same, what do we do then (like they are 2/3 for $p_{00}$ and 1/3 for the rest) – user143234 Apr 15 '14 at 07:32
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This is another question: then one must compute $P(x_n=0)$ first, and that depends on the initial distribution. – Did Apr 15 '14 at 07:33