Saddlepoint Approximation and First-Order Correction Term to the Joint Probability Density Function of M Quadratic and Linear Forms in K Gaussian Random Variables With Arbitrary Means and Covariances

Let w be a Kx1 Gaussian random vector with arbitrary Kx1 mean vector r and KxK covariance matrix R. The general quadratic and linear forms of interest are the M random scalars z(m) = w' P(m) w + p(m)' w +p(m) for m=1 :M, where KxK matrix P(m), Kx1 vector p(m), and scalar q(m) contain arbitrary constants for m=1:M. The joint probability density function (PDF) of Mx1 random vector z=Z(1)... Z(M)' at an arbitrary point in M-dimensional space is desired. An exact expression for the joint moment generating function (MGF) of random vector z is derived. The inability (analytic and numerical) to perform the M-dimensional inverse Laplace transform back to the PDF domain requires use of the saddlepoint approximation (SPA) to obtain useful numerical values for the desired PDF of z. A first-order correction term to the SPA is also employed for more accuracy, which requires fourth-order partial derivatives of the joint cumulate generating function (CGF). Derivation of the fourth-order partial derivatives of the CGF involves some interesting and useful matrix manipulations which are fully developed. Two MATLAB programs for the entire SPA procedure (with correction term) are presented in this report.