The Multivariate Complex Normal Distribution

Abstract

This chapter presents the multivariate complex normal distribution. It is introduced by Wooding (1956), but it is Goodman (1963) who initiates a more thorough study of this area. Furthermore Eaton (1983) describes the distribution by using vector space approach. In this book we have also used vector space approach and the book is the first to give a systematic and wide-ranging presentation of the multivariate complex normal distribution. The results presented are known from the literature or from the real case. First the univariate case is considered. We define the standard normal distribution on ℂ and by means of this an arbitrary normal distribution on ℂ is defined. For the univariate standard complex normal distribution we study the rotation invariance, which says that the univariate standard complex normal distribution is invariant under multiplication by a complex unit. For an arbitrary complex normal distribution the property of reproductivity is examined. This is the characteristic that the sum of complex numbers and independent complex normally distributed random variables multiplied by complex numbers still is complex normally distributed. The normal distribution on ℂ^p is defined and also the reproductivity property for it is studied. For all the distributions the relation to the real normal distribution is determined and the density function and the characteristic function are stated. We also specify independence results in the multivariate complex normal distribution and furthermore marginal and conditional distributions are examined. We investigate some of the results for the complex normal distribution on ℂ^p in matrix form, i.e. for the complex normal distribution on ℂ^n×p.