A class of “central limit theorems” for convolution products of generalized functions

Abstract

Introduction and summary. From the naive viewpoint of a nonprobabilist, the remarkable fact about the central limit theorem is that (under suitable conditions) the average of a large number of random variables converges (in a suitable sense) to a limit which is independenit of the things being averaged, but is completely determined by the averaging process itself. In the present paper, this situation is abstracted from its connection with probability. As we shall see, the restriction to positive functionals which characterizes probability is in no way necessary for validity of the limit theorems. Moreover, we are able to dispense completely with measure theory, in both definitions and proofs. In a purely algebraic and function-space context, we find a whole sequence of averaging processes, each with its own limit function; the first two of them generalize the weak law of large numbers and the central limit theorem. We prove convergence in three different senses, under three different sets of assumptions: first convergence of moments, then uniform convergence of characteristic functions, and finally weak convergence of distribution functions. Our first theorem is purely combinatorial; our next, essentially a calculus theorem; in our third we use function-space techniques borrowed from the modern theory of partial differential equations. By doing so, we obtain rather refined and precise results on the strongest norm in which we can assert convergence. In this respect our results may be of interest even in the classical probabilistic cases. In particular (see Corollary 2 below) we show that if a very mild extra regularity condition is imposed on the summands in the classical central limit theorem, then not only do the partial sums converge in law to the Gaussian, but every derivative of the c.d.f. of the partial sum exists and converges to the corresponding derivative of the Gaussian. Theorem 3 possesses an interpretation in terms of electrical engineering, which is given in the Remark following Corollary 2. It also has a surprising application to numerical analysis, which is described in the Postscript. We begin with an abstract algebra A over the real numbers, equipped with a linear functional Y. (If a E A were a random variable, Y(a`) would be its kth