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The problem of designing optimum memoryless detectors for known signals in long-range dependent (LRD) noise is considered. In particular, under the performance criterion of asymptotic relative efficiency (ARE), optimum memoryless detection in LRD noise is investigated by exploiting the Hermite expansions. The detectors considered have the form of a nonlinearity followed by an accumulator and threshold comparator. It is shown that when the noise is LRD and Gaussian, all nonlinearities with Hermite rank one are asymptotically equivalent in terms of efficiency and are most powerful. Moreover, the optimum nonlinearities with Hermite rank greater than one are given by the corresponding Hermite polynomials. The case of non-Gaussian LRD noise that can be derived by nonlinear transformation of Gaussian noise is also considered. In this case, the globally optimum nonlinearity is very difficult to obtain in general. Instead, we proposed a suboptimal nonlinearity, which is given by a linear combination of the corresponding locally optimum detector and the inverse of the transform function generating the noise. Simulations show that the proposed detector outperforms the locally optimal detector for LRD non-Gaussian noise.