For a random walk say from point $x$ to $y$ on a graph,

How is the probability of a Random walker reaching point $y$ before returning to $x$ related to the expected of the number of visits to point $x$, $E(T_{x})$ and the expected number of visits to y, $E(T_{y})$ after infinitely many steps? (If there does exist one can you please prove it)

  • $\begingroup$ with classical notations, $ET_x$ is the average time to return to $x$ starting from $x$. $\endgroup$ – mookid Mar 26 '14 at 4:35
  • $\begingroup$ yes that's correct $\endgroup$ – user42382 Mar 26 '14 at 4:42
  • $\begingroup$ so, not the expected number of visits. $\endgroup$ – mookid Mar 26 '14 at 4:48
  • $\begingroup$ well i was kinda assuming they're the same $\endgroup$ – user42382 Mar 26 '14 at 5:03
  • $\begingroup$ isn't the number of visits infinite? $\endgroup$ – mookid Mar 26 '14 at 5:03

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