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For the study of the random velocity fluctuation of the solid Earth medium, it is useful to measure the collapse of a seismic wavelet with increasing travel distance and the excitation of coda waves. Radiative transfer theory (RTT) is a powerful tool for synthesizing the propagation of a seismic wavelet in random media statistically characterized by the power spectral density function (PSDF) of the fractional velocity fluctuation. The Born scattering coefficient is a key building block of RTT. As the centre wavenumber of a wavelet increases, the phase shift across the correlation length increases and the Born approximation leads to an extremely large forward scattering exceeding the applicable range of the perturbation method. In such a case, the Eikonal approximation is able to explain the envelope broadening with increasing travel distance; however, it can not explain the coda excitation. To overcome the difficulty, we have proposed a hybrid Monte Carlo (MC) simulation for scalar waves. In the case of von Kármán-type random media, when the centre wavenumber is higher than the corner wavenumber, taking the centre wavenumber as a reference, we divide the PSDF into two spectral components. Applying the Born and Eikonal approximations to the high- and low-wavenumber spectral components, we statistically evaluate the wide-angle scattering and the narrow-angle ray bending, respectively. The proposed MC simulation serially using two kinds of scattering processes successfully synthesizes the time trace of the wave energy density from the onset to the late coda. The travel-distance fluctuation derived from the one-way propagation of the Eikonal approximation is also important. This paper extends this method for the propagation of a vector wavelet in random elastic media. We suppose that random fractional fluctuations of the P - and S -wave velocities and the mass density are linearly proportional to each other, which maintains the linear polarization of an S wave throughout the scattering process. Using the hybrid MC simulation with the spectrum division, we synthesize three-component energy density time traces for the anisotropic radiation from a moment tensor source, from which we derive three-component root mean square (RMS) velocity amplitude time traces at different azimuths. In parallel, we synthesize the propagation of a vector wavelet in many realized random elastic media by the finite-difference simulation, then we calculate three-component RMS velocity amplitude time traces. Using them as a benchmark, we confirm the validity of the proposed MC simulation for specific cases. [ABSTRACT FROM AUTHOR]