Taking one-dimensional random transverse Ising model (RTIM) with the double-Gaussian disorder for example, we investigated the spin autocorrelation function (SAF) and associated spectral density at high temperature by the recursion method. Based on the first twelve recurrants obtained analytically, we have found strong numerical evidence for the long-time tail in the SAF of a single spin. Numerical results indicate that when the standard deviation {sigma}_{JS} (or {sigma}_{BS}) of the exchange couplings J_{i} (or the random transverse fields B_{i}) is small, no long-time tail appears in the SAF. The spin system undergoes a crossover from a central-peak behavior to a collective-mode behavior, which is the dynamical characteristics of RTIM with the bimodal disorder. However, when the standard deviation is large enough, the system exhibits similar dynamics behaviors to those of the RTIM with the Gaussian disorder, i.e., the system exhibits an enhanced central-peak behavior for large {sigma}_{JS} or a disordered behavior for large {sigma}_{BS}. In this instance, the long-time tails in the SAFs appear, i.e., C(t)simt^{-2}. Similar properties are obtained when the random variables (J_{i} or B_{i}) satisfy other distributions such as the double-exponential distribution and the double-uniform distribution.
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