Source: J. Appl. Probab. Volume 41, Number 4
(2004), 1230-1236.
It is well known that the Kelly system of proportional betting,
which maximizes the long-term geometric rate of growth of the
gambler's fortune, minimizes the expected time required to reach a
specified goal. Less well known is the fact that it maximizes the
median of the gambler's fortune. This was pointed out by the
author in a 1988 paper, but only under asymptotic assumptions that
might cause one to question its applicability. Here we show that
the result is true more generally, and argue that this is a
desirable property of the Kelly system.
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription.
Read more about accessing full-text
References
Breiman, L. (1961). Optimal gambling systems for favorable games. In Proc. 4th Berkeley Symp. Math. Statist. Prob., Vol. 1, University of California Press, Berkeley, pp. 65--78.
Mathematical Reviews (MathSciNet):
MR135630
Edelman, D. (1979). Supremum of mean--median differences for the binomial and Poisson distributions: $\ln2$. Tech. Rep., Department of Mathematical Statistics, Columbia University.
Ethier, S. N. (1988). The proportional bettor's fortune. In Gambling Research: Proc. 7th Internat. Conf. Gambling Risk Taking, Vol. 4, ed. W. R. Eadington, Bureau of Business and Economic Research, Reno, NV, pp. 375--383.
Gnedenko, B. V. and Kolmogorov, A. N. (1968). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, MA.
Mathematical Reviews (MathSciNet):
MR233400
Hall, P. (1980). On the limiting behaviour of the mode and median of a sum of independent random variables. Ann. Prob. 8, 419--430.
Mathematical Reviews (MathSciNet):
MR573283
Hamza, K. (1995). The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions. Statist. Prob. Lett. 23, 21--25.
Kelly, J. L. Jr. (1956). A new interpretation of information rate. Bell System Tech. J. 35, 917--926.
Mathematical Reviews (MathSciNet):
MR90494
Leib, J. E. (2000). Limitations on Kelly, or the ubiquitous `$n\to\infty$'. In Finding the Edge: Mathematical Analysis of Casino Games, eds O. Vancura, J. A. Cornelius and W. R. Eadington, Institute for the Study of Gambling and Commercial Gaming, Reno, NV, pp. 233--258.
Maslov, S. and Zhang, Y.-C. (1998). Optimal investment strategy for risky assets. Internat. J. Theoret. Appl. Finance 1, pp. 377--387.
Thorp, E. O. (2000). The Kelly criterion in blackjack, sports betting, and the stock market. In Finding the Edge: Mathematical Analysis of Casino Games, eds O. Vancura, J. A. Cornelius and W. R. Eadington, Institute for the Study of Gambling and Commercial Gaming, Reno, NV, pp. 163--213.
