A method is described for constructing approximations, to statistical functions, that are uniformly convergent in time, starting with the expansion of the functions as Taylor series in time. The principal tool is a technique for expanding the Fourier transform of the unknown function by use of a set of orthonormal functions. Application to the Lagrangian velocity correlation and the eddy diffusivity for marked particles in a three-dimensional random velocity field yields results that agree excellently with computer simulations. The approximation procedure is extended to expansions in strength parameters (e.g. Reynolds number expansion) and to an expansion about the direct-interaction approximation. The latter is based on a new model representation of the direct-interaction approximation. An implication of the work is that the usual diagram expansions, obtained through term-by-term averaging over a Gaussian distribution, may not uniquely determine the functions they represent; it may be that truly meaningful expansions are possible, in general, only for distributions which bound the amplitudes in the individual realizations.