It is a simple matter to produce an random number, and by stacking k 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 Z 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 k random numbers, stacking them, and performing the transformation