On Optimal Detectors in Multi-User Detection Problems

The paper is a supplement to [1]. Conditions under which asymptotically optimal detectors are linear are found. It is shown also that if, in contrast to [1], we consider not the Bayesian but minimax statement of the problem with unknown coefficients, then optimal detectors are linear (moreover, nonasymptotically). A geometrical meaning of Theorem 1 from [1] is explained, and it is shown that the theorem follows from some general results [2, 3] on hypotheses testing. It is also shown that some results of [4] follow from [1, Theorem 1].