Abstract
The concept of pattern arises in many applications comprising experimental trials with two or more possible outcomes in each trial. A binary scan of type r / k is a special pattern referring to success–failure strings of fixed length k that contain at least r-successes, where r, k are positive integers with \(r\le k\). The multiple scan statistic \(W_{t,k,r}\) is defined as the enumerating random variable for the overlapping moving windows occurring until trial t which include a scan of type r / k. In the present work, we consider a sequence of independent binary trials with not necessarily equal probabilities of success and develop upper bounds for the probability of the event that the multiple scan statistic will perform a jump from \(\ell \) to \(\ell +1\) (where \(\ell \) is a nonnegative integer) in a finite time horizon.
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Acknowledgements
D.P.L. would like to dedicate this work in memory of his father Panagiotis (Takis). His support during this research endeavour of D.P.L. is one of the many moving memories which the co-author will always recall, full of love and gratitude! The authors would like to thank the anonymous reviewers for their helpful and constructive comments that contributed to improving the original version of this paper. 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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Koutras, M.V., Lyberopoulos, D.P. Asymptotic results for jump probabilities associated to the multiple scan statistic. Ann Inst Stat Math 70, 951–968 (2018). https://doi.org/10.1007/s10463-017-0621-1
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DOI: https://doi.org/10.1007/s10463-017-0621-1