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A Generalization of the Classical Kelly Betting Formula to the Case of Temporal Correlation

For sequential betting games, Kelly's theory, aimed at maximization of the logarithmic growth of one's account value, involves optimization of the so-called betting fraction $K$. In this Letter, we extend the classical formulation to allow for temporal correlation among bets. To demonstrate the potential of this new paradigm, for simplicity of exposition, we mainly address the case of a coin-flipping game with even-money payoff. To this end, we solve a problem with memory depth $m$. By this, we mean that the outcomes of coin flips are no longer assumed to be i.i.d.random variables. Instead, the probability of heads on flip $k$ depends on previous flips $k-1,k-2,...,k-m$. For the simplest case of $n$ flips, with $m = 1$, we obtain a closed form solution $K_n$ for the optimal betting fraction. This generalizes the classical result for the memoryless case. That is, instead of fraction $K^* = 2p-1$ which pervades the literature for a coin with probability of heads $p\geq 1/2$, our new fraction $K_n$ depends on both $n$ and the parameters associated with the temporal correlation. Generalizations of these results for $m > 1$ and numerical simulations are also included. Finally, we indicate how the theory extends to time-varying feedback and alternative payoff distributions.