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

Abstract

Let x/sup n/ denote a sequence built over a finite alphabet A, and let P(x/sup n/;w) be the probability of x/sup n/ generated by the source w. We define a uniquely decodable code /spl phi/(x/sup n/) of length |/spl phi/(x/sup n/)|=-logQ(x/sup n/) where Q(/spl middot/) is an arbitrary probability distribution on A/sup n/. The cumulative redundancy of the encoding x/sup n/ at the output of a source w is defined as p(x/sup n/;/spl phi//sub n/,w):=-logQ(x/sup n/)+logP(x/sup n/). Finally, let us consider a set of sources /spl Omega/, and define the minimax redundancy as p/sub n/(/spl Omega/):=inf/sub /spl phi/n/sup/sub w/spl isin//spl Omega//max/sub xn/spl isin/An/{p(x/sup n/;/spl phi//sub n/,w)}. We study asymptotically /spl rho//sub n/(/spl Omega/) for memoryless sources via analytic methods.