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A presentation of relationships for the conditional probability distributions of the stage at which tests of sequential hypotheses terminate. These probability distributions permit study of sequential tests in which the boundaries are functions of the sample number. The authors 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 the method apply in evaluating various ad hoc schemes for truncating tests and to the theory of tests with a varying parameter. (See also RM-4338-PR.)

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