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April, 1988 Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment
Paul H. Algoet, Thomas M. Cover
Ann. Probab. 16(2): 876-898 (April, 1988). DOI: 10.1214/aop/1176991793


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 $k \rightarrow \infty$.


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Paul H. Algoet. Thomas M. Cover. "Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment." Ann. Probab. 16 (2) 876 - 898, April, 1988.


Published: April, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0642.90016
MathSciNet: MR929084
Digital Object Identifier: 10.1214/aop/1176991793

Primary: 90A09
Secondary: 28D20 , 49A50 , 60F15 , 60G40 , 94A15

Keywords: asymptotic equipartition property (AEP) , asymptotic optimality principle , capital growth rate , ergodic stock market , expected log return , gambling , log-optimum portfolio , Portfolio theory , Shannon-McMillan-Breiman theory

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 2 • April, 1988
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