The describing-function method for the analysis of nonlinear systems with sinusoidal inputs is interpreted as a mean-square quasi-linearization technique and is generalized to apply to random signals. An amplitude-sensitive (or zero-memory) nonlinearity is interpreted as being approximately equivalent to a linear frequency-insensitive device, and a formula is derived for its equivalent gain. A simple rate-limited control system with a Gaussian input is analyzed as a specific application. Approximation of a general nonlinear element (containing memory) is considered next, and a relation between equivalent-system impulse response and the response of the actual nonlinearity is derived.