Abstract
A random Rayleigh vector is a vector of norms of random Gaussian vectors, and a Chi-square vector is the vector of squares of these norms. Excepting special cases, neither the Rayleigh nor Chi-square distribution function is known in closed form. The present study examines limiting forms of their distribution functions with a view towards providing useful approximations. In particular it is shown that, as the noncentrality parameters become large, the limiting form of both the m-dimensional Rayleigh and Chi-square distributions is m-dimensional Gaussian. Furthermore, asymptotic bounds are given for the error of the Gaussian approximation to the m-dimensional Chi-square distribution. Some additional properties of these distributions are given as well.
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