Previous calculations of the statistical behavior of a hard spring oscillator excited by purely random and Gaussian noise are extended. The moments of the displacement and its extrema are found in an analytic form and their interrelations are exploited. Recurrence relations, asymptotic expressions, and computations of the moments are presented. Envelope autocorrelations for three narrow‐band linear filters are found, and the mean size of clumps of cycles exceeding predetermined levels is obtained. For the hard‐spring oscillator we find that the relative envelope fluctuation is less than that of a linear oscillator and that the mean clump size is also smaller in general.