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The Probability Theory of Patterns and Runs has had a long and distinguished history, starting with the work of de Moivre in the 18th century and that of von Mises in the early 1920's, and continuing with the renewal-theoretic results in Feller's classic text An Introduction to Probability Theory and its Applications, Volume 1. It is worthwhile to note, in particular, that de Moivre, in the third edition of The Doctrine of Chances (1756, reprinted by Chelsea in 1967, pp. 254-259), provides the generating function for the waiting time for the appearance of k consecutive successes. During the 1940's, statisticians such as Mood, Wolfowitz, David and Mosteller studied the distribution theory, both exact and asymptotic, of run-related statistics, thereby laying the foundation for several exact run tests. In the last two decades or so, the theory has seen an impressive re-emergence, primarily due to important developments in Molecular Biology, but also due to related research thrusts in Reliability Theory, Distribution Theory, Combinatorics, and Statistics.
Sommario
1. Combinatorial Probability.- 1. Expressions in Terms of Binomial Coefficients for some Multivariate Distributions of Order k.- 2. Success Runs in a Circular Sequence of Independent Bernoulli Trials.- 3. Inversion Techniques and Combinatorial Identities: A Quick Introduction to Hypergeometric Evaluations.- 4. The Two-Color Whitworth Problem for Long Cycles or Long Lines.- 5. Binomial Distributions of Order k on the Circle.- 6. Length of the Longest Non-Decreasing Subsequence on Two Symbols.- 7. Exchangeability and Recursion in the Conditional Distribution Theory of Number and Length of Runs.- 8. The Longest Run of any Letter in a Randomly Generated Word.- 2. Reliability Theory and Statistics.- 9. Applications of Consecutive System Reliability in Selecting Acceptance Sampling Strategies.- 10. Exact and Approximate Hypercube Reliabilities.- 11. Runs and Patterns with Applications to Reliability.- 12. Robust Procedures for Detecting Non-Random Patterns.- 13. Reliability of Consecutive Weighted k-out-of-n:F System.- 3. Limit Theorems and Approximations.- 14. Pattern Recognition in Markov Chains.- 15. On the Waiting Time of Appearance of Given Patterns.- 16. Runs and Excursions.- 17. Applications of the Stein-Chen Method to the Theory of Patterns and Runs: An Annotated Bibliography.- 18. Runs of Superimposed Poisson Processess.- 19. Circular Overlapping Success Runs.- 20. Asymptotic Joint Distribution of Cover Times.
