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Let T be a discrete time random variable describing the time of first visit for a MC from an initial state i and returning back to state i of a state space $S={0,1}$.

Let $P\left[T \geq n\right]$ be the cumulative probability distribution.

By definition, this is $\sum_{n_{i} \leq n}P\left[n_{i}\right]=P\left[1\right]+ \cdot \cdot \cdot P\left[n-1\right]$

But, how is the above equals to

$P\left[X_{n=1}=1, \cdot \cdot \cdot, X_{n-1}=1 | X_{0}=0\right]$?

Is there a theorem or definition that I have missed out?

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    $\begingroup$ What on earth is "Let $P[T\leq n$] be the cumulative probability distribution of the random variable $T$ less than or equal to $n$" supposed to mean? $\endgroup$ – Math1000 May 8 '18 at 14:19
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If the first return time for state 0 is $\geq n$ then you have to stay at state 1 from time 1 up to time $n-1$.

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