Showing papers on "Complex normal distribution published in 1973"

Hermite–gaussian functions of complex argument as optical-beam eigenfunctions

Abstract: Optical-resonator modes and optical-beam-propagation problems have been conventionally analyzed using as the basis set the hermite–gaussian eigenfunctions ψn (x,z) consisting of a hermite polynomial of real argument Hn [√2x/w(z)] times the complex gaussian function exp [−jkx2/2q(z)], in which q(z) is a complex quantity. This note shows that an alternative and in some ways more-elegant set of eigensolutions to the same basic wave equation is a hermite-gaussian set ψˆn(x,z) of the form Hn[√cx]exp [−cx2], in which the hermite polynomial and the gaussian function now have the same complex argument √cx ≡ (jk/2q)1/2x. The conventional functions ψn are orthogonal in x in the usual fashion. The new eigenfunctions ψˆn, however, are not solutions of a hermitian operator in x and hence form a biorthogonal set with a conjugate set of functions ϕˆn(√cx). The new eigenfunctions ψˆn are not by themselves eigenfunctions of conventional spherical-mirror optical resonators, because the wave fronts of the ψˆn functions are not spherical for n > 1. However, they may still be useful as a basis set for other optical resonator and beam-propagation problems.

On a Minimax Estimate for the Mean of a Normal Random Vector Under a Generalized Quadratic Loss Function

Abstract: An admissible minimax estimate for the mean of a normal random vector with known covariance is derived for a generalized quadratic loss function. This loss function is quadratic in both the estimation error and the unknown mean. The estimate is derived using the method of least favorable prior distributions. The decision rule is linear, and the least favorable prior distribution for the unknown mean is normal with zero mean. The covariance of this least favorable normal distribution is determined by the solution of a certain nonlinear algebraic matrix equation.

On Strengthening Lyapunov Type Estimates (The Case When the Distribution of the Summands is Close to the Normal Distribution)

Abstract: We agree to denote by L and b, with or without subscripts, absolute positive constants. The problem of obtaining estimates for the quantity 6 in which the righthand sides tend to zero as the distributions of the summands approximate the normal distribution was apparently first considered in [1]. (Earlier in [2], the case was investigated where they become close to stable laws.) In particular, for a pair of distributions (F, G) [1] and [2] introduced the quantity

On a test for reality of the covariance matrix in a complex gaussian distribution

Abstract: Goodman ( 1963 ), Wooding ( 1956 ), James ( 1967 ) and Khatri ( 1965a, 1965b ) have studied some distribution problems of the complex ,P-variate normal distribution which appears in time series analysis and physics. Most problems of complex variates can be treated in the same way as those of the real variates in the case of normal distributions. Khatri ( 1965a ) has derived the likelihood-ratio criterion for testing the independence of the two sets of variates ( see section 2 ) of a complex matrix variate with complex Gaussian distribution. In this paper, we derive the distribution of the criterion, when the hypothesis is true, for P = 4, 5, 6 and 7 and also give the general form of the same for any P. Tables of correction factors for converting chi-square percentiles of a logarithmic function of the criterion are obtained for P = 4 and 5 and for selected values of n -q