Let $(X_n)_{n \geq0}$ be a Markov Chain on {$0,1,...$} with transition probabilities given by: $P_{0,0}=1, \ \ P_{i,i+1}= p_i, \ \ P_{i,i-1}= q_i, \ $ for all $i \geq 1$, and $P_{i,j}=0 \ $ otherwise.

Let $T_i = \text{min}${$n \ge 0 \ : \ X_n=i$}.

I was reading through a problem in my text with the above information given and found that the fact$\ \lim_{m \to \infty} \mathbb{P}(T_0 < \infty|X_0 = m) = 0 \ $, was used in a proof but there was no justification as to why it was true. Intuitively this makes sense because as your initial state becomes larger and larger, each 'path' to state $0$ has a lower and lower probability of happening. But I can't seem to prove this rigorously.


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