I'm an economist looking at Markov Chains for the first time, and this is simple stuff and it is quite embarassing to post about this, because the answer is provided, but I do not understand the solution to the problem below. Can someone shed some lights on this (the exercise is in the link)?
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What exactly do you not understand? A basic theorem in the theory of Markov chains is that $(X_n)$ is Markov on a countable state space $S$ with transition matrix with entries $p_{ij}$ and initial distribution $\pi$ if and only if for any $N+1$ states, $i_0,\dotsc,i_N$, the following equation holds: $$P(X_0=i_0,\dotsc, X_N=i_N)=\pi_{i_0}p_{i_0 i_1} \cdot \dotso \cdot p_{i_{N-1} i_N},$$ which is exactly what they state and use to calculate the probability of a given path of states. – Nap D. Lover Dec 30 '17 at 18:51
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Hi, yeah, I can see it now. It's blindingly obvious, actually. Either lack of sleep or a dumb moment... – Fede C Dec 31 '17 at 05:06