The detectability of random signals in the presence of noise

This morning you have heard excellent presentations of two fields of endeavor, the results and techniques of which could be basic to a statistical theory of communication engineering. On the one hand, the field of statistical inference, as applied to discrete stochastic processes, has developed to a refined point due to the efforts of many statisticians. — The work of the late Professor Wald in his successful application of Von Neumann's game theory to the construction of a general theory of decision functions has played a dominant role in the development of these refinements. On the other hand, the theory of stochastic processes depending upon a discrete or continuous time parameter has been developed during the last three deoades by various mathematicians, only during the last few years has the study of statistical inference problems for continuous stochastic processes received much attention. Here the outstanding contribution is the thesis of Ulf Grenander, published in the Arkiv för matematik, Band 1, Häfte 3, 1950. In attempting to apply the techniques of statistical inference to continuous processes, it is evident that the central problem is to obtain a coordinate system for the process which allows one to actually carry out. the computations called for by various statistical methods. As far as I am aware, there are at present only two types of continuous stochastic processes for which a coordinate system has been obtained with which one can carry through some of the computations necessary in the testing of statistical hypotheses. One process is a projection on to the real axis of a finite dimensional Markoff process, Gaussian or non Gaussian. The other process is a Gaussian process with a continuous covariance function. The restriction of a continuous covariance function is not serious, since this property applies to all of the stochastic models which have been set up to study continuous processes occurring in communication engineering. (The assumption that the spectrum of a process is a "pure white noise" is not consistent with continuity of the covariance function, but a pure white noise is merely a mathematical idealization. The process with a flat band limited spectrum — a model often used in application — does possess a continuous covariance function.) On the other hand, the restriction to Gaussian processes is one which would be desirable to remove in some cases.