It is supposed that there is a multichannel sensor system which performs sequential detection of a target. Sequential detection is done by implementing a generalized Wald's sequential probability ratio test, which is based on the maximum-likelihood ratio statistic and allows one to fix the false-alarm rate and the rate of missed detections at specified levels. We present the asymptotic performance of this sequential detection procedure and show that it is asymptotically optimal in the sense of minimizing the expected sample size when the probabilities of erroneous decisions are small. We do not assume that the observations are independent and identically distributed (i.i.d.). The first-order asymptotic optimality result holds for general statistical models that are not confined to the restrictive i.i.d. assumption. However, for i.i.d. and quasi-i.i.d. cases, where the log-likelihood ratios can be represented in the form of sums of random walks and slowly changing sequences, we obtain much stronger results. Specifically, using the nonlinear renewal theory we are able to obtain both tight expressions for the error probabilities and higher order approximations for the average sample size up to a vanishing term. The performance of the multichannel sequential detection algorithm is illustrated by an example of detection of a deterministic signal in correlated (colored) Gaussian noise. In this example, we provide both the results of theoretical analysis and the results of a Monte Carlo experiment. These results allow us to conclude that the use of the sequential detection algorithm substantially reduces the required resources of the system compared to the best nonsequential algorithm.