The entropy of a randomly stopped sequence

A Wald-like equation is proved for the entropy of a randomly stopped sequence of independent identically distributed discrete random variables X₁, X₂. . ., with a nonanticipating stopping time N. The authors first define a general stopping time and the associated stopped sequence, and then present the two main theorems for the entropy of a stopped sequence. The formal proofs of the lemmas necessary for the proof of the theorems are given. The randomness in the stopped sequence X^N is the expected number of calls for X times the entropy per call plus the residual randomness in the stopping time conditioned on the unstopped sequence X^∞