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Sequential analysis has been used in the literature for analyzing a radar search system. The procedure is to continually sample a given range "bin" until a yes-no decision concerning the presence of a target is made. The results, however, do not provide for the possibility of a target emerging into the bin during the sampling process. In this report we consider a likelihood-ratio test for detecting emerging targets by sequential sampling. It is shown that the decisioning threshold must be continually increased in order to maintain a fixed false alarm rate at each sample. An upper bound on the average number of samples required to detect an emerging target is computed, and the results are significant in that this bound may become quite large. A test using a fixed threshold in which the average false alarm rate is constrained is investigated and shown to be inferior to the variable threshold test. A suboptimum test using only a finite number of past samples is also considered. The results of this study are equally applicable to the converse problem, i.e., detecting when a known target exits from the bin, which is important in target tracking.