Demimartingale Approaches for Scan Statistics
Scan statistics are defined as random variables enumerating the moving windows in a sequence of binary outcome trials which contain a prescribed number of successes. The main objective of this contribution is to serve as a self-contained source of some recent results concerning both the simple and the multiple scan statistic. These results are innovative in the sense that they seem to be the first ones on scan statistics that were derived by means of demimartingale techniques. The demimartingale approach motivated also some classification questions for stochastic processes associated with scan statistics. These types of questions and some past results on scan statistics that can be regarded as relevant to the demimartingale approach are also discussed here. In order to illustrate how our results can be implemented in practice, our presentation is enriched with several numerical exhibitions.
KeywordsScan statistic Multiple scan statistic Demimartingale Demisubmartingale N-demimartingale N-demisupermartingale Bound Asymptotic results Maximal inequalities Binary trial
D.P.L. would like to dedicate this work in memory of his father Panagiotis (Takis). His support during this research endeavor of D.P.L. is one of the many moving memories which the co-author will always recall, full of love and gratitude!
This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) – Research Funding Program: Aristeia II - Investing in knowledge society through the European Social Fund.
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