Through a careful examination of the equations by which Wald determines the values of the boundaries for tests of sequential hypotheses, we are able to obtain interesting relationships for the conditional probability distributions of the stage at which the test terminates. These enable us to study sequential tests in which the boundaries are functions of the sample number. We are particularly concerned with tests with convergent boundaries, and investigate a set of boundaries which approach those of a truncated Wald test. Approximate expressions for the expected sample number and probabilities of error are obtained for the tests considered. The obtained approximations apply best for gently tapering slopes. Extensions of our method can be applied to evaluate various ad hoc schemes for truncating tests and to the theory of tests with a varying parameter.

In this paper the techniques of sequential analysis are applied to a multiple-resolution-element search radar for different types of detection. By using sequential methods such a radar shows considerable saving compared to fixed-scan-rate detection radars. Many of the problems occurring in sequential radar detection are examined; the difficulties of obtaining analytical results for multiple-resolution-element sequential radars are discussed. Most of the results of this paper were obtained by simulation. Included are interesting results which suggest that truncation is not necessary for the tests which were considered. Curves are given describing the improvement obtained by sequential detection procedures. The curves are discussed in detail to make this paper useful to those whose primary interest is in the results which are obtained.

The "point of regeneration" method is used to obtain simple sequential equations for determining the complete probability density function for multiple occurrences of events in a Bernoulli sequence. Both the independent and overlapping classes of recurrent events are included in the general framework of these equations. The equations also lead to the generating function for the probability distribution. This is used to obtain the expected recurrence times for the different classes of recurrent events. A distinction is made between the probability distributions for the occurrence of thekth event at thenth trial and the occurrence ofkevents inntrials. The latter case is of primary concern. The methods employed and the results obtained have extensive applications in problems in automatic control, communications, and information processing.