Contents

• Contents
• Introduction
  • Outline and Summary
  • Notation and Concepts
• Modelling with Markov Chains
  • Markov Chains
  • Coupled-, Overlapping- and Higher-Order Chains
    • Coupled Chains
    • Overlapping Chains
    • Higher Order Chains
  • Spectral Analysis of Markov Chains
    • Perron's Formula
    • Frequency Analysis
    • The Central Limit Theorem
• Serial Tests and Markov Chains
  • Serial Tests for Pseudorandom Sequences
    • The case s=1
    • The case s>1 with non-overlapping tuples
    • The case s>1 with overlapping tuples
  • Test Statistics for Frequency Analysis
    • Pearson's Statistic for Multinomial Variates
    • Quadratic Forms in Weak Inverses
• A Generalized $\phi$-Divergence
  • From Pearson's Statistic to the $\varphi$-divergence
  • Generalizing the Quadratic Form
  • The $\tilde{I}_{\overline{\Sigma},\varphi}$-Divergence
  • Backward Compatibility
• Examples
  • Modelling and Testing Bohr's Hydrogen Atom
    • From States to Transitions
    • Dimension Reduction
  • Testing Long Period Generators
    • The Gambling Test
    • Empirical Results
• Appendix
  • Lemmata
  • Proof of the Asymptotic Distribution of $I_{\overline {\Sigma },\phi }$
  • Expectation and Covariance Matrix of $\tilde{C}^{(n)}$
  • A Weak Inverse for Multinomial Distributions
• Bibliography