The design of asymptotically optimum detectors (AODs) is considered for a class of fluctuating targets in non-Gaussian interference, as a reasonable alternative to optimum (Neyman-Pearson) detectors, which prove hopelessly complex. The useful signal is modelled as an incoherent train with constant amplitude or with fluctuating amplitude, either from scan to scan or from pulse to pulse. The AOD structure is based on an asymptotical expression of the log-likelihood ratio, which has been derived by a suitable modification of the classical AOD approach; it results in a conventional detector (CD) but for the replacement of the quadratic envelope detection with another zero-memory nonlinearity, depending only on the noise statistics. The invariance properties of the CD with respect to the signal fluctuation model are shown to be preserved in the more general AOD class. As the test statistics are asymptotically normal, the asymptotical AOD operating characteristics are expressed in terms of asymptotical relative efficiency (ARE), thus allowing direct comparison with the CD under the same interference. The ARE for log-normal interference is calculated, showing a potential improvement on the order of at least 20 dB.