A Note on the Summability of the Entropy Series

where p~ log p,~ is defined as 0 whenever Pn = 0 [see, for example, Feinstein (1958)]. The function H~ is a sum of nonnegative terms and consequently invariant under rearrangements of the terms. The probabilities p,~ may therefore be reordered so that they are monotonic decreasing (nonincreasing), to facilitate discussion of the finiteness of H , . All terms may also be assumed positive, for nontriviality. We hereafter confine attention to monotonic nonincreasing sequences of positive probabilities p = { p ~ ) . (Actually, ultimate monotonicity is enough.) In this note we present two sufficient conditions to assure the finiteness of the entropy H~ for such distributions p. They are given in the two theorems below. The proofs are based on corollaries to Lemmas 1 and 2, respectively; these lemmas and corollaries may have some independent interest. That H~ need not be finite may be seen from the example