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Getting Back to Zero II

In the previous post I showed that a return to zero is certain when playing a fair game with an unbiased coin. So if you're in the hole you would think that just holding on long enough will get you out, or at least back to zero. This may be a bit overoptimistic. What follows is a closer look at getting back to zero in the case of a fair game.

For a fair coin the first return probability generating function is \(F(z) = 1 - \sqrt{1-z^2}\). \(\quad F(1)=1\) so the first return probabilities constitute a proper probability distribution, and a return to zero at some time or another is certain. To find the mean number of tosses or games required to get back to zero, you take the derivative of \(F(z)\) and evaluate it at \(z=1\). The derivative is

\[\acute{F}(z) = \frac{z}{\sqrt{1-z^2}}\]

At \(z=1\) the derivative is \(\infty\). This means that even though a return to zero is certain, you may have to wait a very long time for it to happen. To be fairly certain of getting out of the hole you will need a very large bankroll and be willing to play for a very long time.

This result can be made more precise. Using a result from the theory of random walks you can show that the probability of not getting back to zero in the first \(2N\) games is equal to the probability of getting to zero at game \(2N\). This means that if \(\bar{R}_{2N}\) is the probability that a return to zero does not occur in the first \(2N\) games then

\[\bar{R}_{2N} = g_{2N} = \binom{2N}{N}/2^{2N}\]

The surprising thing is how long it takes for this probability to become negligible. After \(50\) games the probability is \(0.1123\), and after \(100\) it is still \(0.07959\). Even after \(1000\) games the probability of not returning to zero is still \(0.02522\). (Note that I have left out the derivation of this result but if you are interested, leave a comment and I will post it.)

The conclusion is that even for a fair game it may take a very long time for you to get out of a hole. On the other hand if you are ahead then it is possible that you may hold on to your lead for a very long time. Maybe this explains the apparent success of some money managers?

© 2010-2012 Stefan Hollos and Richard Hollos

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