The sequential estimation and detection of signals in normal noise. II

The object of this paper is to present results on the sequential detection of known signals, and of signals known except for unknown parameters, when gaussian noise is present. The principal analytical tool for the study is the Karhunen-Loéve expansion of a random process in terms of the characteristic functions of the covariance kernel. If the process is continuous in the mean, the expansion converges in mean square to the original process over the interval of definition (the observation interval). The well-known results on this expansion all relate to a fixed observation interval. When the length of the observation interval is allowed to vary, as in the case of sequential analysis, some further properties of the expansion must be derived as a preliminary to an attack on the statistical problems. These properties, which might be considered as results in probability theory, are presented in Part I of the present paper,1 along with a statement of the problems to be studied in a form suitable for the sequel. Part II presents more special results of a statistical nature.1See preceding paper