...device
If the domain of application is known one might wish to use more specialized generation methods like (t,m,s)-nets. For the latter notion, see Niederreiter [38]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...iud
i.e. independent uniformly distributed 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...quickly
By information we mean the ability to distinguish distributions. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...thesis
This question would involve physics (e.g. quantum mechanics) as well as philosophic discussions. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...estimators
Estimators again express the frequentists' interpretation of the meaning of probability since they are usually built on relative frequency. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...'measurement'
e.g. estimation 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...himself
A. Kolmogorov, On Tables of Random Numbers, Sankhya Ser. A, p.369, 1963, quoted from [13, p.94,] 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...events
We refer the reader to [32] for an introduction to complexity theory and to [5] for the notion of randomness with respect to finite strings. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...events
The theory can also be build without explicitly referring to a sample space. The important object is always the sigma field because it represents the events of interest. In this thesis we will always assume that a sample space is given and that the random variable is defined for all elements in the sample space. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...all
with respect to the according probability space 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...logical
i.e. statements within the axiomatic setup that do not depend on the subjective relation to reality that we have defined in Definition 1.2. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...numbers
However, the class of sequences of random variables leading to laws of large numbers is much larger. Consider for example ergodic transformations that can lead to dependent random variables, but still suffice to prove laws of large numbers. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...plate
A short discussion on the importance of such red plates is given by Ian Steward in [44] 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...randomness
By 'randomness' we denote the difficulty of predicting the outcomes of such devices. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...sense
See [32] for an introduction to complexity theory. When defining randomness as absence of information, periodic behavior clearly cannot be considered random! 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...processes
Consider the ergodic transformation 4 x (1-x) which can be viewn as discretization of the continuous equation which describes the growth of a population of bacteria under some limiting conditions. 

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...[0,per-1]
per stands for 'maximal possible period length' and expresses the fact that all known methods produce periodic sequences of integers. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...generators
We refer the reader to L'Ecuyer [28, Section 3.6,] and to Niederreiter [38]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...'random'
Again, randomness is understood in the sense of numerical properties relevant for the application. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...made
Consider the difference between the distribution of balls drawn from an urn with and without replacement. See also L'Ecuyer [28, p.5,] 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...LCGs
We would like to thank Hannes Leeb of the PLAB-group for the permission to reproduce the following five plots. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...properties
Statistical properties are properties of distributions. RNs have empirical distributions. Empirical distributions are numerical properties of a set of real numbers whereas distributions are assumptions on a model. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...discretizised
Discretization of a domain is a risky step in forming a model for a real world phenomena since it can lead to unexpected dynamics. However, this is just an introductory example. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...regularities
Consider lattices, long range correlations, or periodicity for example. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...'number'
See Leeb [30, p.16,] for a proof of this statement. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...high
That is, 145#145 can be made arbitrarily near to zero. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...generator
Recent results have shown new techniques to overcome this problem. They do not make assertions on one generator, but calculate the arithmetic mean of the test statistic when a large set of generators is applied. We refer the reader to [35], [12] and [11]. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...properties
in the sense of passing a certain test 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...however
See [34, 35] for the estimation of the star-discrepancy. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...same
This has been proved by Leeb in [30, p.28ff,] 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...Koksma
The inequality can be generalized to higher dimensions and to other sets of functions. For an introduction we refer the reader to [36, Chapter 2,]. 

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...ways
See also Section 3.4 in Chapter 3 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...equal
e.g. better suited to certain applications, 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...length
In this chapter we use 216#216 for the number of vectors that are tested. In the other chapters, N denotes the number of (onedimensional) random numbers. These are related by the equation 217#217 in the case of nonoverlapping tuples and by 218#218 for overlapping tuples. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...independent
If two sets of random variables have the property that every random variable in the first set is independent from every RV in the second set, so will be two functions defined on only the first and second set respectively. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...size
Here we always assume that the sequence 4#4 is a finite leading segment of an infinite sequence of independent uniformly distributed RVs. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...all
with respect to the measure P 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...number
Do not confuse this 145#145 with the significance level of a statistical test! 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...multiplying
Of course, this will be done without carrying out the multiplication, but by type conversion from the type float to the type long integer in C, or appropriate types in other languages. The computer has not got anything like real numbers, and the mentioned 32 bits are actually present. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...decide
in the statistical sense 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...results
Consider the weighted spectral test, Hellekalek [19], for example. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...test
M stands for the dimension of the tuples and thus has the same meaning as s in the 31#31 goodness-of-fit test. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...first
In this case 216#216 equals N. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...too
This and also further guidelines for reporting results in connection with computer based statistical testing can be found in [23]. 

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...methods
These have been described in Chapter 3 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...good
that is, small 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...do
In an application that treats the PRNs as real numbers the user has to decide whether later bits will influence the results of the stochastic model. We have already mentioned the importance of such considerations. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...generator
See Altman [1] for a detailed discussion. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...seconds
The time statistic was measured on a DEC 3000 alpha workstation. 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

Stefan Wegenkittl
Tue Dec 3 09:56:35 MET 1996