Second- and Third-Order Asymptotics of the Continuous-Time Poisson Channel

The paper derives the optimal second-order coding rate for the continuous-time Poisson channel. We also obtain bounds on the third-order coding rate. 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 induced by Wyner's discretized channel and the construction of an appropriate $\epsilon$-net of the input probability simplex. 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.