Second- and Third-Order Asymptotics of the Continuous-Time Poisson Channel
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The paper derives the optimal second- and third-order coding rates for the continuous-time Poisson channel. This is the first instance of a second-order result for a continuous-time channel. The converse proof hinges on a novel construction of an output distribution consisting of a convex combination of countably infinitely many distributions indexed by a subset of the rationals. While the achievability proof follows the general program to prove the third-order term for non-singular discrete memoryless channels put forth by Polyanskiy, several non-standard techniques---such as new definitions and bounds on the probabilities of typical sets using logarithmic Sobolev inequalities---are employed to handle the continuous nature of the channel.
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