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

## TL;DR

This paper establishes the optimal second-order coding rate for the continuous-time Poisson channel and provides bounds on the third-order rate, marking the first such result for a continuous-time channel.

## Contribution

It introduces the first second-order asymptotics for a continuous-time channel and develops novel techniques for the converse and achievability proofs.

## Key findings

- Optimal second-order coding rate derived
- Bounds on third-order coding rate obtained
- First second-order result for a continuous-time channel

## Abstract

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.