Cauchy Distribution - Multivariate Cauchy Distribution

Multivariate Cauchy Distribution

A random vector X = (X1, ..., Xk)′ is said to have the multivariate Cauchy distribution if every linear combination of its components Y = a1X1 + ... + akXk has a Cauchy distribution. That is, for any constant vector a ∈ Rk, the random variable Y = a′X should have a univariate Cauchy distribution. The characteristic function of a multivariate Cauchy distribution is given by:

where x0(t) and γ(t) are real functions with x0(t) a homogeneous function of degree one and γ(t) a positive homogeneous function of degree one. More formally:

and for all t.

An example of a bivariate Cauchy distribution can be given by:

Note that in this example, even though there is no analogue to a covariance matrix, x and y are not statistically independent.

Analogously to the univariate density, the multidimensional Cauchy density also relates to the multivariate Student distribution. They are equivalent when the degrees of freedom parameter is equal to one. The density of a k dimension Student distribution with one degree of freedom becomes:

f( {\mathbf x} ; {\mathbf\mu},{\mathbf\Sigma}, k)= \frac{\Gamma\left}{\Gamma(1/2)\pi^{k/2}\left|{\mathbf\Sigma}\right|^{1/2}\left^{(1+k)/2}} .

Properties and details for this density can be obtained by taking it as a particular case of the multivariate Student density.

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