The dynamic response of stiffness degrading models subjected to random excitations is investigated. Monte Carlo simulation is performed to determine the effect on the response of various system parameters and to provide results for testing the applicability of approximate analytical techniques. The principal focus is on the single degree of freedom Clough model for reinforced concrete structures with white excitations lasting eight nominal periods of the structure. A heuristic equivalent linearization method is proposed to estimate the response of degrading structures through approximate analysis. The equivalent linear stiffness and damping at any time are computed from certain weighted averages of the secant stiffness and the hysteretic energy dissipation associated with limiting hysteresis loops corresponding to the maximum amplitude at that time. The drift component of the displacement is not treated explicitly, but is approximately accounted for by assigning appropriate weighting functions in this scheme. The equivalent linear coefficients are expressed as functions of some measure of the maximum response statistics and are modified at specified increments of time. The problem of predicting such maximum response statistics is also investigated for the simpler process representing the nonstationary response of linear structures. An approximate method is developed and compared with two existing approximate methods. The new method is found to yield reasonably accurate estimates of the mean and the variance of the response maximum with significant improvements in computational efficiency as compared to simulation. The applicability of a closure approach for finding the response of degrading models is also investigated. A modified Gaussian closure procedure is shown to yield good approximations for two simpler problems which embody some aspects of the Clough model: the maximum value problem for linear systems, and the random response of bilinear hysteretic structures. For the Clough model also, this technique has been applied successfully.