We construct a family of random step functions\{x_n (t)\}whose members all have the same power spectrum and such that asn \rightarrow \infty , x_n (t)converges tox_{\infty} (t), the Gaussian process with the same spectrum. We illustrate the procedure for calculating the general multivariate distribution of the processes\{x_n(t)\}by calculating the univariate, bivariate and trivariate distributions. We show how a suitably constructed univariate entropy can serve as an index of the extent to whichx_n(t)has approached the Gaussian limitx_{\infty}( t).

It is shown how to construct shot noise which, like the turbulent velocity field, is quite accurately univariate normal, but exhibits marked departures from bivariate normality at close ranges.

We study the power spectrum and, more generally, the spectral covariance of weighted stationary processes. It is found that if the power spectrum of the underlying stationary process is suitably well behaved and properly matched to the weight function, then the high-frequency behavior of the power spectrum and spectral covariance is especially simple. Asymptotic theorems describing this behavior precisely are given.

The chi process is defined as a natural generalization of the chi distribution of statistical theory. A formula is derived for the expected number of level crossings per second of the chi process. The formula contains as a special case the familiar expression for the fluctuation rate of the envelope of Gaussian noise.

A new kind of random process, the locally stationary random process, is defined, which includes the stationary random process as a special case. Numerous examples of locally stationary random processes are exhibited. By the generalized spectral density\Psi(\omega, \omega \prime)of a random process is meant the two-dimensional Fourier transform of the covariance of the process; as is well known, in the case of stationary processes,\Psi(\omega, \omega \prime)reduces to a positive mass distribution on the line\omega = \omega \primein the\omega, \omega \primeplane, a fact which is the gist of the familiar Wiener-Khintchine relations. In the case of locally stationary random processes, a relation is found between the covariance and the spectral density which constitutes a natural generalization of the Wiener-Khintchine relations.

A detailed analysis of the general binary channel is given, with special reference to capacity (both separately and in cascade), input and output symbol distributions, and probability of error. The infinite number of binary channels with the same capacity lie on double-branched equicapacity lines. Of the channels on the lower branch of a given equicapacity line, the symmetric channel has the smallest probability of error and the largest capacity in cascade, unless the capacity is small, in which case the asymmetric channel (with one noiseless symbol) has the smallest probability of error and the largest capacity in cascade. By simply reversing the designation of the output (or input) symbols, we can decrease the probability of error of any channel on the upper branch of the equi-capacity line and increase the capacity in cascade of any asymmetric channel on the upper branch. In a binary channel neither symbol should be transmitted with a probability lying outside the interval[1/e, 1 - (1/e)]if capacity is to be achieved. The maximally asymmetric input symbol distributions are approached by certain low-capacity channels. For these channels, redundancy coding permits an appreciable fraction of capacity in cascade if sufficient delay can be tolerated.