$P_{i,1}=\frac{i}{i+1}$ and $P_{i,i+1}=\frac{i}{i+1},i\in\mathbb{Z}$
Can someone walk me through how to prove this DTMC Random Walk scenario is either transient, null recurrent or positive recurrent? I have read countless posts and different notes in textbooks and cannot figure out how to go about properly showing this. I simply start by i=1 and notice a pattern and try and disprove it being positive recurrent then prove transient or null recurrent but I am truly struggling..
after cleaning up some open typos, I infer that you'll have probabilities $p_k\in(0,1)$ $P :=\left(\begin{matrix} 1-p_1 & p_1 & 0 & 0 & 0 & 0 & \dots\\ 1-p_2 & 0 & p_2& 0 & 0 & 0 & \dots\\ 1-p_3 & 0 & 0 & p_3 & 0 & 0 & \dots\\ 1-p_4 & 0 & 0 & 0 & p_4 & 0 & \dots\\ 1-p_5 & 0 & 0 & 0 & 0 & p_5 & \dots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots\end{matrix}\right)$
which is known as the (renewal) Age Matrix and is discussed in some detail in Feller vol 1 (3rd ed) Markov Chains chapter.
By inspection there is a single communicating class.
To verify transience/ recurrence, you merely need to check for state 1. And you don't return WP1 to state one iff
$\prod_{k=1}^\infty p_k \gt 0$.
Sums are easier to evaluate than products, so use the below identity, with $\delta_k = 1-p_k$
$\prod_{k=1}^\infty (1-1/2^k)$ converge to zero?
If the chain is transient, then you are done. If the chain is recurrent, then you may evaluate positive recurrence by simply checking whether the mean time until revisiting state 1 is finite. This may be computed directly as
$E\big[T\big] = \sum_{k=0}^\infty P\big(T\gt k\big) = 1 + \sum_{k=1}^\infty \prod_{j=1}^k p_j$
