he following is the program of our
actions:
We start from the uniform on
random variables
and apply the BoxMuller procedure, described in the claim
(
BoxMuller procedure
). The step
is a unitary linear transformation of the iid standard normal variables
into a jointly normal variables
.
The transformation
produces two correlated uniform variables according to the
(
Sklar_theorem_2
). The transformation
is the application of the (
Sklar_theorem_1
).
We spell out every step below.
We choose the transformation
,
to construct the
with the following
properties
We observe
that
Hence, it suffices to chose
according
to
The step
is performed according to the idea behind the result
(
Sklar_theorem_2
). We note that the cumulative
standard normal distribution
produces a uniform on [0,1] random variable
from a standard normal variable
according to the rule
Indeed, for such
we calculate
We also introduce the function
and uniform on [0,1] random variables
:
Hence,
For the final step
we are given the cumulative distributions
.
We simulate the variables
and
according to the
rules
Such variables have marginal cumulative distributions
and
respectively. Their correlation is controlled by the parameter
.
The joined distribution is given by the following
calculation:
If
is 0 then the
splits into product and the
are uncorrelated. Preservation of sign of correlation takes place.
