To overcome these problems, the concept of pseudorandom numbers has been proposed: simply substitute the physical device by a computer algorithm that generates numbers in a deterministic, fast and reproducible way. Since the mechanism generating pseudorandomness is now open to mathematical analysis, numerical qualities can be guaranteed in principle. These so-called theoretical tests are complemented by empirical tests that should assess qualities for which a mathematical analysis of the algorithm cannot be done (yet).
We now are able to give a first definition of what is understood by PRNs.
The reader may compare this definition with that given by Ripley [43, p.15,].
We have adopted the term 'numerical properties' instead of 'statistical properties' since we
have already seen that any definition of pseudorandomness using the
term 'distribution' or
'statistical' will cause conceptual difficulties in showing the desired properties for a specified
sequence of PRNs. We can only calculate the probability assigned to the PRN by a certain
model X. But this reflects a property of both, the model and the PRN, and cannot be
considered a good definition for the PRN itself.
Moreover, computer algorithms can only produce periodic
sequences due to their finite state space. These sequences
can theoretically be shown to be nonrandom in a certain
sense
.
We have to start in the limited world of algorithms in order to characterize the pseudorandom numbers in a way that is both mathematically precise and practically relevant. Forget the idea of real numbers, forget distributions, forget the notion of limits. And forget the rest of this thesis because it will illustrate what has been done since the first of these PRNs have been produced by an algorithm, deliberate of the conscience of the definition given above or not: showing that the PRNs have 'statistical' properties that give reason to treat this numbers as realizations of random variables with a given distribution.
We will try to introduce the lie on which both theoretical and empirical testing (when understood in the sense of giving recommendation for the use of certain PRN sequences) is built. Notice the usage of the terms random number and pseudorandom number in this thesis. One difference between PRNs and RNs is that in the former case we know the deterministic character of the numbers, whereas this is left open to philosophical discussion in the latter case. Since we have formulated the relation between probability and reality in terms excluding the relevance of such discussions, we will not have to distinguish between PRNs and RNs due to this criterion. What remains is the following: Firstly the resolution of the numbers and secondly the notion of 'relevance for an application'.
As to the first point, RNs are real numbers defined as limits of sequences of
rational numbers. PRNs are rational numbers. In most cases the resolution of the PRNs will
be fine enough to treat them as RNs. Attention has to be paid to the effects of
discretizations. The according considerations have to be
made by
anyone who uses computer simulation.
Since the discovery that chaos may emerge from the
discretization of non-chaotic continuous processes
this warning should not be overlooked.
The second point is even more important: whenever we use the term PRN we stress the relevance for a specific application. Random numbers are innocent whereas PRNs always can be judged with respect to an application. Roughly speaking PRNs are random numbers yielding the desired results in an application.
In order to understand the mentioned lie we have to examine the notion of the empirical distribution function.