## Follows now from Theorem 5.2. Q. E. D

Note that this result holds regardless of whether the matrix BE BT is nonsingular or not. In the latter case the normal distribution involved is called “singular”:

Definition 5.2: Ann x 1 random vector Y has a singular Nn (д, E) distribution if its characteristic function is of the form yY (t) = exp(i ■ t т д – 21TE t) with E a singular, positive semidefinite matrix.

Because of the latter condition the distribution of the random vector Y involved is no longer absolutely continuous, but the form of the characteristic function is the same as in the nonsingular case – and that is all that matters. For example, let n = 2 and

*=й -=(0 "0.

where a 2 > 0 but small. The density of the corresponding N2(*, —) distribution

of Y = (Yb Y2)t is

Then lima4...

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