Cumulants are useful in studying nonlinear phenomena and in developing (approximate) statistical properties of quantities computed from random process data. Wavelet analysis is a powerful tool for the approximation and estimation of curves and surfaces. This work considers both wavelets and cumulants, developing some sampling properties of linear wavelet fits to a signal in the presence of additive stationary noise via the calculus of cumulants. Of some concern is the construction of approximate confidence bounds around a fit. Some extensions to spatial processes, irregularly observed processes and long memory processes are indicated.