  In classic Kelly gambling, bets are chosen to maximize the expected log
growth of wealth, under a known probability distribution. Breiman provides
rigorous mathematical proofs that Kelly strategy maximizes the rate of asset
growth (asymptotically maximal magnitude property), which is thought of as the
principal justification for selecting expected logarithmic utility as the guide
to portfolio selection. Despite very nice theoretical properties, the classic
Kelly strategy is rarely used in practical portfolio allocation directly due to
practically unavoidable uncertainty. In this paper we consider the
distributional robust version of the Kelly gambling problem, in which the
probability distribution is not known, but lies in a given set of possible
distributions. The bet is chosen to maximize the worst-case (smallest) expected
log growth among the distributions in the given set.
  Computationally, this distributional robust Kelly gambling problem is convex,
but in general need not be tractable. We show that it can be tractably solved
in a number of useful cases when there is a finite number of outcomes with
standard tools from disciplined convex programming.
  Theoretically, in sequential decision making with varying distribution within
a given uncertainty set, we prove that distributional robust Kelly strategy
asymptotically maximizes the worst-case rate of asset growth, and dominants any
other essentially different strategy by magnitude. Our results extends
Breiman's theoretical result and justifies that the distributional robust Kelly
strategy is the optimal strategy in the long-run for practical betting with
uncertainty.
