On the asymptotics of the minimax redundancy arising in a universal coding

Let xⁿ denote a sequence built over a finite alphabet A, and let P(xⁿ;w) be the probability of xⁿ generated by the source w. We define a uniquely decodable code φ(x ⁿ) of length |φ(xⁿ)|=-logQ(xⁿ) where Q(·) is an arbitrary probability distribution on Aⁿ . The cumulative redundancy of the encoding xⁿ at the output of a source w is defined as p(xⁿ;φ_n,w):=-logQ(xⁿ)+logP(xⁿ ). Finally, let us consider a set of sources Ω, and define the minimax redundancy as p_n(Ω):=inf_φnsup_w∈Ω max_xn∈An{p(xⁿ;φ_n,w)}. We study asymptotically ρ_n(Ω) for memoryless sources via analytic methods