Volume of the space of positive definite sequences

Abstract

The coefficients or a normalized real positive definite sequence, such as a normalized autocorrelation sequence, are, constrained to be less than or equal to unity in magnitude. It is well known that the coefficients are further constrained, but the constraints become complicated rather quickly as the sequence order increases. One measure of the additional constraints placed on positive definite sequences is the volume of the portion of real space they occupy. The ratio of that volume to the volume of the hypercube circumscribing the positive definite region can be interpreted as the probability that a random sequence restricted to less than unity in magnitude is in fact a positive definite sequence. The author derives a closed-form expression for the volume of the space occupied by positive definite sequences of any order. The derivation provides a simple iterative procedure to determine the constraints placed on each autocorrelation coefficient in terms of lower-order coefficients. As a byproduct of the derivation, further insight is provided into the high spectral sensitivity of autocorrelation coefficients to small perturbations near the positive definite boundary. >