  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.
