It is a simple matter to produce an
random number, and by stacking
such numbers in a column vector you obtain a vector with independent standard normal components
. The meaning of the terms independence and randomness in the context of a deterministic algorithm required for the generation of these numbers is somewhat subtle; see Knuth (1973, Vol. 2, Chapter 3) for a discussion of these issues.
Rather than
, what is required is the generation of a vector
—that is,
with covariance matrix
where
If the covariance matrix is symmetric and positive definite, it has a Cholesky root
such that
can be factored as
where
is lower triangular. See Ralston and Rabinowitz (1978, Chapter 9, Section 3-3) for details. This vector
can be generated by the transformation
. Note that this is where the assumption of multivariate normality is crucial. If
, then
is also normal or Gaussian. The mean of
is
and the variance is
Finally, let
; that is, you add a mean term to each variable
. The covariance structure of the
remains the same. Unconditional simulation is done by simply repeatedly generating
random numbers, stacking them, and performing the transformation
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