I'm taking a stochastic processes class, and we looked at the example of Gambler's ruin with infinite target, i.e. the gambler stops when he reaches 0 fortune or N, in the limit of N going to infinity.

For a finite target, in the case of $p=q$, the probability to ruin and expected time to ruin for starting fortune $a$ are given by:

$$P_a = 1 - \frac{a}{N}$$

$$T_a = a(N-a)$$

As $N\rightarrow \infty$, $P_a \rightarrow 1$ and $T_a \rightarrow \infty$. Now there seems to be something very wrong with the probability measure on the set of possible trajectories. In calculating the probability to ruin, the set of trajectories going to infinity is of measure zero, but in calculating the stopping time, it seems that the set of trajectories going to infinity have been assigned some positive measure. So what is going wrong when taking the limit $N\rightarrow \infty$?

When N is finite, the number of possible trajectories is countably infinite, is this still the case in the limit $N\rightarrow \infty$?

  • $\begingroup$ What is the "stopping time" in this context ? The time when the gambler is ruined ? $\endgroup$ – Peter Mar 17 '18 at 17:52
  • $\begingroup$ Yes, time to reach zero fortune $\endgroup$ – The Hagen Mar 17 '18 at 17:54
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    $\begingroup$ A similar situation occurs in throwing a fair dice until the number of "heads" and "tails" coincide. The probability that this happens eventually is $1$, but the expected time for the occurence is $\infty$ $\endgroup$ – Peter Mar 17 '18 at 17:55
  • $\begingroup$ You should have $P_a = 1-\frac aN$, since $\frac aN$ is the probability of reaching $N$ starting from $a$. $\endgroup$ – Math1000 Mar 19 '18 at 8:21
  • $\begingroup$ Corrected, thank you! $\endgroup$ – The Hagen Mar 19 '18 at 15:55

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