In Section 1.4.1 we have stated that a sequence of random numbers does not have anything like a distribution. However, everybody in the field of stochastic simulation speaks of sequences of PRNs having this or that distribution and we have to take a closer look at the meaning of such a statement.
A sequence of independent uniform (P)RNs should be a sequence of numbers that behaves somewhat typical for a sequence of independent uniformly distributed random variables. In [26, p.3,], Knuth has stated that PRNs should appear to be random. In this sense, iud PRNs should appear to be independent and uniformly distributed.
Now given a (finite) sequence of pseudorandom numbers
, where 
we have to show that
behave like realizations of a random variable distributed uniformly in [0,1[,
behave like independent realizations of such a random variable.
Since an iud sequence of random variables can be defined by using the multidimensional
uniform distribution we may restate
the two requirements in a single definition:
However, this definition remains incomplete until we have clarified
what is understood by 'behaves like a realization':
such numerical properties of the PRNs
will have to be developed with respect to the actual application.
If we do not mark out such properties any further, the ridiculous, but mathematical correct
conclusion is: every possible s-tuple in 
behaves like a realization of an s-dimensional uniformly distributed random vector
since it falls into the range of . Thus
every possible sequence of numbers in [0,1[ has the same right to be called a sequence
of
random numbers. Note that this argument is built on the definition of random numbers and
does not account for the fact that the special model 'uniform distribution' assigns the
same probability to every sequence of PRNs. The argument is thus valid for any desired
distribution. It reflects the difficulty to relate random variables to random numbers.
We have derived this trivial conclusion in a complicated way in order to allow a slight modification, that introduces the numerical properties relevant for an application: since we have a tool for estimating the distribution of a function of random variables, namely the empirical distribution function, why don't we substitute 'has an empirical distribution function near to the s-dimensional uniform distribution' for 'behaves like a realization' in Definition 1.8?
This is the key to every sort of statistical inference on PRNs. The usual arguments run
the following way: fix a function g=g(X) of a random variable X distributed uniformly
on [0,1[. Define that a sequence 
is 'independent uniformly
distributed' if the empirical distribution function 
is near
to
the distribution function 
of g. As we will see in Chapter
2, this amounts to carrying out a statistical test on the sequence of
PRNs.
What is wrong with this definition? It can be criticized in two ways:
but may fail
the test for another function 
. This again stresses the fact that PRNs should be
chosen with respect to their application.
Thus statistical inference on PRNs will lead to numerical properties which cannot be used to judge on the PRNs in terms of 'distribution' or 'randomness'. A statement like 'My generator has passed the RUN-test for randomness' does not tell us that the PRNs from the generator are more random than any other sequence of real numbers. We will further examine and develop these kind of numerical properties in Chapter 2 where we will also give examples.