The response of an arbitrary discretized system to the random movement has been solved in probabilistic terms. The excitation has been defined as a combination of the time modulated band
limited stationary random processes approximating the evolutionary power spectra of a true seismic record. The solution is based either on the modified version of the stochastic Newmark method or on the spectral differential decomposition of the excitation. Special attention has been paid to the applicability of the methods to the sparse problems, especially to their matrix-free formulation.