The problem is considered of designing memoryless detection systems for detecting random signals in qb-mixing noise environments. Assuming that the bivariate densities of the noise can be expressed in an infinite series expansion, the nonlinearity is expressed as a series expansion and the coefficients are found that maximize the asymptotic relative efficiency.

The problem of minimum mean-squared error prediction of a discrete-time random process using a nonlinear filter consisting of a zero-memory nonlinearity followed by a linear filter is studied. Classes of random processes for which the best predictor is realizable using a nonlinear filter of the above form are discussed. For those random processes for which the best predictor is not realizable using the above nonlinear filter, an iterative procedure is presented for finding a suboptimal nonlinear filter; special attention is directed to the case where the nonlinearity is a polynomial. Also, a noniterative approach based on nonlinear regression is presented.

Two classes of random processes that exhibit one-step predictors with optimal nonlinear minimum mean-squared error (MMSE) are discussed, and conditions for membership to one of these classes are given. Examples of each class are presented, and the optimal one-step predictors are given.