Renyi Radius and Sphere Packing Bound: Augustin's Approach

First, Renyi radius is defined for any positive order and for any set of probability measures on any measurable space. ErvenHarremoes conjecture is proved for any set of probability measures with finite Renyi radius. Finiteness of Renyi radius for a positive order α is shown to imply three continuity results: the continuity of Renyi radius as a function of order on (0, α], the uniform equicontinuity of Renyi information as a family of functions of the order on any compact subset of (0, α) indexed by the priors, and the uniform equicontinuity of Renyi information as a family of functions of the prior indexed by the orders in [0, α]. In addition a number of other new results, and some old ones, about Renyi quantities are derived. Second, the channel coding problem reviewed in an abstract framework. If the scaled Renyi radius of a sequence of channels converge to a finite continuous function on an interval of orders of the form (1 − ε, 1] for an ε > 0, then the capacity of the sequence of channels is proved to be equal to the scaled order one Renyi radius. If the convergence holds on an interval of the form (1 − ε, 1 + ε) then the strong converse is proved to hold. Both hypothesis hold for large class of product channels and for certain memoryless Poisson channels. Using Augustin’s approach a sphere packing bound with a polynomial multiplicative approximation error, i.e. prefactor, is established for the decay rate of the error probability with block length for any sequence of product channels {W[1,n]}n∈Z+ for which maxt≤n C 1 2 ,Wt is O(ln n). For stationary discrete product channels with feedback, i.e. discrete memoryless channels with feedback, sphere packing exponent is proved to bound the exponential decay rate of the error probability with block length from above. The latter result continues to hold for product channels with feedback satisfying a milder stationarity hypothesis.