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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 &#x03F5;-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.

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