Universal schemes for sequential decision from individual data sequences

Sequential decision algorithms are investigated in relation to a family of additive performance criteria for individual data sequences. Simple universal sequential schemes are known, under certain conditions, to approach optimality uniformly as fast as n^-1 log n, where n is the sample size. For the case of finite-alphabet observations, the class of schemes that can be implemented by finite-state machines (FSMs) is studied. It is shown that Markovian machines with sufficiently long memory exist, which are asymptotically nearly as good as any given deterministic or randomized FSM for the purpose of sequential decision. For the continuous-valued observation case, a useful class of parametric schemes is discussed with special attention to the recursive least squares algorithm