Coding theorems on the worst-case redundancy of fixed-length coding for a general source

We consider a situation where n-tuples generated from a general source are encoded by a fixed-length code and discuss coding theorems on the worst-case redundancy, where the worst-case redundancy is defined as the maximum of the difference between the rate and the ideal codeword length per symbol with respect to all the correctly decodable n-tuples. We treat the four cases where the decoding error probability ε_n is required to satisfy (a) lim_n→∞ ε_n = 0, (b) lim inf_n→∞ ε_n = 0, (c) lim sup_n→∞ ε_n ≤ ε, and (d) lim inf_n→∞ ε_n ≤ ε, respectively, where ε ∈ [0; 1) is an arbitrary constant. We give general formulas of the optimum worst-case redundancy that are closely related to the width of the entropy-spectrum of a source.