The Annals of Probability

Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment

Paul H. Algoet and Thomas M. Cover

Full-text: Open access


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 $k \rightarrow \infty$.

Article information

Ann. Probab., Volume 16, Number 2 (1988), 876-898.

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 90A09
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.

Export citation