The computational cost of estimating normalized correlations can be reduced by employing sums of simple nonlinear functions of the data. A quadruplex transformation is presented, and the performance of the associated estimator is analyzed for complex Gaussian processes. With independent observations, the variance of the estimator is approximately 14% higher than that of the conventional estimator.

Computers and brains are modeled by finite and probabilistic automata, respectively. Probabilistic automata are known to be strictly more powerful than finite automata. The observation that the environment affects behavior of both computer and brain is made. Automata are then modeled in an environment. Theorem 1 shows that useful environmental models are those which are infinite sets. A probabilistic structure is placed on the environment set. Theorem 2 compares the behavior of finite (deterministic) and probabilistic automata in random environments. Several interpretations of Theorem 2 are discussed which offer some insight into some mathematical limits of machine intelligence.

Using kernel estimates of the Parzen type, a naive sequential nonparametric density estimation procedure is developed. The asymptotic distribution structure of the stopping variable is examined. The stopping variable is shown to have finite moments of ail order and is shown to be dosed. The stopping variableNdepends on some preassigned error\varepsilon, and it is shown thatNdiverges strongly to\inftyas\varepsilonconverges to zero. Finally, with\hat{f}_n(x)being a kernel-type estimator, it is shown that\hat{f}_N(X)converges tof(x), the true density atx, with probability one as\varepsilonconverges to zero.