A Result on Multidimensional Quantization

For any integer N > I , a probability space, a Gaussian random vector X defined on the space with a positive definite covariance matrix, and an JV-level quantizer Q are presented such that the random vector Q{X) takes on each of the N values in its range with equal probability and such that X and Q{X) are independent. Introduction Quantization, the process by which a set is mapped into a finite subset of a given cardinality, plays a pivotal role in virtually any application that requires analog to digital conversion; indeed, it is at the heart of much of modern digital technology. In such applications, a quantizer is often taken to be a function mapping Rk into a subset of Rk of cardinality N, where A: is a positive integer and N is an integer greater than one (see, e.g., [1, 5, 3, 6, 2, 7, 9]). In this paper we present what might be a surprising consequence of such a general approach to quantization. Development For a topological space T, we will let £&(T) denote the family of Borel subsets of T. For a set S, we will let P(5') denote the power set of S and Is denote the indicator function of S. By a standard Gaussian measure we will mean a Gaussian measure whose first moment is zero and whose second moment is one. Let k be a positive integer. For any measure m on (Rk, &(Rk)) we will let ra* denote the inner measure on (Rk , P(R/c)) induced by m and we will let m* denote the outer measure on (R*, P(Rfc)) induced by m. Recall from [4, p. 61] that if B e &(Rk) and A e P(Rk), then m,(B n A) + m*(B n Ac) = m(B). We will let k denote Lebesgue measure on (R, <^(R)) and A denote Lebesgue measure on (Rk , &(Rk)) for integers k > 1, where k will be determined from the context. Recall that for a measure space Received by the editors April 20, 1991 and, in revised form, September 17, 1991; a preliminary version of this paper was presented at the 863rd meeting of the American Mathematical Society. 1991 Mathematics Subject Classification. Primary 60A10; Secondary 62H12.