The Annals of Probability
- Ann. Probab.
- Volume 16, Number 2 (1988), 876-898.
Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment
We ask how an investor (with knowledge of the past) should distribute his funds over various investment opportunities to maximize the growth rate of his compounded capital. Breiman (1961) answered this question when the stock returns for successive periods are independent, identically distributed random vectors. We prove that maximizing conditionally expected log return given currently available information at each stage is asymptotically optimum, with no restrictions on the distribution of the market process. If the market is stationary ergodic, then the maximum capital growth rate is shown to be a constant almost surely equal to the maximum expected log return given the infinite past. Indeed, log-optimum investment policies that at time $n$ look at the $n$-past are sandwiched in asymptotic growth rate between policies that look at only the $k$-past and those that look at the infinite past, and the sandwich closes as $k \rightarrow \infty$.
Ann. Probab., Volume 16, Number 2 (1988), 876-898.
First available in Project Euclid: 19 April 2007
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Secondary: 94A15: Information theory, general [See also 62B10, 81P94] 28D20: Entropy and other invariants 60F15: Strong theorems 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 49A50
Portfolio theory gambling expected log return log-optimum portfolio capital growth rate ergodic stock market asymptotic optimality principle asymptotic equipartition property (AEP) Shannon-McMillan-Breiman theory
Algoet, Paul H.; Cover, Thomas M. Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment. Ann. Probab. 16 (1988), no. 2, 876--898. doi:10.1214/aop/1176991793. https://projecteuclid.org/euclid.aop/1176991793