For gambling on horses, a one-parameter family of utility functions is proposed, which contains Kellys logarithmic criterion and the expected-return criterion as special cases. The strategies that maximize the utility function are derived, and the co
nnection to the Renyi divergence is shown. Optimal strategies are also derived when the gambler has some side information; this setting leads to a novel conditional Renyi divergence.
The achievable error-exponent pairs for the type I and type II errors are characterized in a hypothesis testing setup where the observation consists of independent and identically distributed samples from either a known joint probability distribution
or an unknown product distribution. The empirical mutual information test, the Hoeffding test, and the generalized likelihood-ratio test are all shown to be asymptotically optimal. An expression based on a Renyi measure of dependence is shown to be the Fenchel biconjugate of the error-exponent function obtained by fixing one error exponent and optimizing the other. An example is provided where the error-exponent function is not convex and thus not equal to its Fenchel biconjugate.
The rate region of the task-encoding problem for two correlated sources is characterized using a novel parametric family of dependence measures. The converse uses a new expression for the $rho$-th moment of the list size, which is derived using the relative $alpha$-entropy.
Two families of dependence measures between random variables are introduced. They are based on the Renyi divergence of order $alpha$ and the relative $alpha$-entropy, respectively, and both dependence measures reduce to Shannons mutual information wh
en their order $alpha$ is one. The first measure shares many properties with the mutual information, including the data-processing inequality, and can be related to the optimal error exponents in composite hypothesis testing. The second measure does not satisfy the data-processing inequality, but appears naturally in the context of distributed task encoding.
