The effects of randomization on finite-memory decision schemes

This paper is concerned with the differences between deterministic and randomized finite-memory decision rules. It is shown that for any hypothesis-testing problem there exists a
b < \infty
such that, for any
B
, the optimal deterministic rule with
B + b
bits in memory has a lower error probability than the optimal randomized rule with
B
bits in memory. Suboptimal deterministic rules with this property are demonstrated. These deterministic rules lose at most
b
bits. Thus for large memories the fraction of memory, measured in bits, lost by deterministic rules is negligible.