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This paper considers the single antenna, static Gaussian broadcast channel inthe finite blocklength regime. Second order achievable and converse rateregions are presented. Both a global reliability requirement and per-userreliability requirements are considered. The two-user case is analyzed indetail, and generalizations to the $K$-user case are also discussed. Thelargest second order achievable region presented here requires bothsuperposition and rate splitting in the code construction, as opposed to the(infinite blocklength, first order) capacity region which does not require ratesplitting. Indeed, the finite blocklength penalty causes superposition alone tounder-perform other coding techniques in some parts of the region. In thetwo-user case with per-user reliability requirements, the capacity achievingsuperposition coding order (with the codeword of the user with the smallest SNRas cloud center) does not necessarily gives the largest second order region.Instead, the message of the user with the smallest point-to-point second ordercapacity should be encoded in the cloud center in order to obtain the largestsecond order region for the proposed scheme.

On Second Order Rate Regions for the Static Scalar Gaussian Broadcast Channel