Non-Asymptotic Bounds of Cumulant Generating Function of Codeword Lengths in Variable-Length Lossy Compression

Shota Saito, Toshiyasu Matsushima

Abstract: This paper investigates the problem of variable-length source coding with the criteria of the normalized cumulant generating function of codeword lengths and the excess distortion probability. We analyze the non-asymptotic fundamental limit of the normalized cumulant generating function of codeword lengths under the constraint that the excess distortion probability is allowed up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon \in [0,1)$ </tex-math></inline-formula> . Our non-asymptotic achievability and converse bounds are characterized by the quantity related to the Rényi entropy.