The Quadratic Gaussian One-Help-Two Source-Coding Problem with Markovity

We consider the quadratic Gaussian one-help-two source-coding problem with Markovity, in which three encoders separately encode the components of a memoryless vector-Gaussian source that form a Markov chain and the central decoder aims to reproduce the first and the second components in the chain subject to individual distortion constraints. For this problem, we determine the minimum sum rate of the first and the second encoder given the distortion constraints and the rate of the third encoder. In particular, a simple scheme consisting of vector quantization followed by Slepian-Wolf binning achieves this minimum sum-rate. The proof of the converse draws from the quadratic Gaussian two-encoder source-coding problem, the Gaussian scalar-help-vector source-coding problem, and the Gaussian many-help-one source-coding problem.