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In the case of a simple random walk $\{S_n, n \ge 0\}$ what is $S_n$. I see this for $P\{S_n=i |\ |S_n|=i_{n-1},...,|S_1|=i_1 \} = \frac{p^i}{p^i+q^i}$

What does this mean? is it: the probability of going to state i after n steps (not sure about that) is $\frac{p^i}{p^i+q^i}$ , which is independent of n.

knk
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A random walk, in the context of Markov chains, is often defined as $S_n = \sum_{k=1}^n X_k$ where $X_i$'s are usually independent identically distributed random variables. My understanding of your given statement is the probability of the summation $S_n$ reaching value $i$ given all its previous history.

Ali
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  • I am referring to the Ross book and he calls $|S_n|$ the absolute distance of the simple random walk from origin. So this means we are at state i after starting at origin with probability $\frac{p^i}{p^i+q^i}$? – knk Nov 18 '13 at 05:08
  • Suppose that you are at origin and start moving away at discrete random increments. What I understand from your statement is that the probability of being at distance $i$ from the origin at the $n$-th stage of your walk given that you have been at distance $i-1$ in the $n-1$-th stage is given as $\frac{p^i}{p^i+q^i}$. – Ali Nov 18 '13 at 05:24