For a random walk, let $a$ denote the probability that the markov chain will ever return to state $0$ given that it is currently in state $1$. Because the markov chain will always increase by $1$ with probability $p$ or decrease by $1$ with probability $1-p$ no matter what its current state, note that $a$ is also the probability that the markov chain currently in the state $i$ will ever enter state $i-1$, for any $i$.

I cannot see how the bold sentence can be true, could anyone help me to understand it? Thanks very much.

  • $\begingroup$ Thanks for asking this very good question. Something that is at the moment also not entirely clear to me. The argument that is given feels right, but I can't see it. I am now looking in some material I have about markov chains and try to figure your question out. It will also make things more clear for me. $\endgroup$ – Pedro Mar 26 '15 at 0:56
  • $\begingroup$ This relies on the property of your specific randomwalk being that regardless what state you are currently in, you will increase your state by one with probability $p$ and decrease with probability $(1-p)$. Now... what is so special about states 1 and 0? It extends infinitely in either direction, so there are no walls to run into. What would happen if I relabeled everything by shifting all of the numbers on the states to the right by $i-1$ without changing your point of focus? Is there any reason to expect the probability to have changed? $\endgroup$ – JMoravitz Mar 26 '15 at 0:59
  • $\begingroup$ I am trying to see how we can calculate that value $a$ in a simple random walk. $\endgroup$ – Pedro Mar 26 '15 at 1:00
  • $\begingroup$ Also I think it makes a difference whether $p$ is bigger or smaller than $0.5$ for the case on infinity. If you go to infinity and your probability of going to the next state is bigger than $0.5$, the probability that you will ever return to that state will decrease and decrease and eventually become $0$ I guess, because you are falling deeper and deeper in the markov chain. $\endgroup$ – Pedro Mar 26 '15 at 1:03
  • $\begingroup$ My main question now is, how is $a$ calculated. $\endgroup$ – Pedro Mar 26 '15 at 1:04

The probability of going ever back to a previous state $a$ given that you are in state $s_i$, depends on the transition probabilities (both upward and downward transition probabilities) between all subsequent states $s_j$ and $s_{j+1}$ with $j \geq i$ and the probability of going from state $i$ to state $i-1$.

As an example, let us say you start in state $1$, the different ways that you can reach state $0$ are you go immediatly from state $1$ to state $0$, you can go to state $0$ by going first from state $1$ to state $2$ and then boing back to state $1$ and then going to state $0$, ... . There exist an infinite amount of ways of going from state $1$ to state $0$. Note that all those different possibilities of going to the previous state are all very similar (they use the same transition probabilities) and that in both cases you need to sum up all those different combinations (an infinite sum, see further) that will give you as a result the probability $a$ you are looking for.

Let us view an example in which we depicted to starting states $s_1$ and $s_3$ and we want to know what the probability is that we go back to state $s_0$ and state $s_2$ respectively.

enter image description here

It is obvious that only the transitions in the picture at the right of our current state $s_1$ (top chain in the figure) respectively $s_3$ matter (bottom chain in the figure). The transitions that matter are given the color red.

Both red series of probabilities that matter are entirely the same sum, if we sum them up to infinity. That is why the probability that we ever go back to the previous state are the same in both cases. Both series of probabilities $\sum_{j=1}^{\infty} P(\tau_{j})$ with $\tau_j$ the event representing the probability that you get in the previous state in $j$ steps, will be the same.

A different scenario would have happened when these probabilities would not have been everywhere the same. Then there would be differences in the probability of ever going back to your previous state.

  • $\begingroup$ Sorry, I still did not see what is the relation between the conclusion and your explanation. $\endgroup$ – Jakoer Mar 26 '15 at 0:40
  • $\begingroup$ I think I got it completely, thanks for you guys' discussion, its really helpful. @JMoravitz $\endgroup$ – Jakoer Mar 26 '15 at 1:41
  • $\begingroup$ You are really a strict and enthusiastic guy, I learn much from you, not just the question, thanks again.I think I would be much better if I studied with you from year 1, ha~ $\endgroup$ – Jakoer Mar 26 '15 at 1:50

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