Large Deviation Behavior for the Longest Head Run in an IID Bernoulli Sequence
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Abstract
Let \(S_N\) be the length of the longest 1-run in an \(N\)-length sequence of Bernoulli trials with parameter \(p\). The famous Erdős-Rényi Law tells that \({S_N}/{\ln N}\rightarrow \xi (p)\) almost surely as \(N\rightarrow \infty \). In this paper, by deriving a sharp lower bound on \(\mathbb{P }(S_N<k)\) for \(N\ge 4k\) and \(k\) large, the deviation probabilities \(\mathbb{P }(S_N/\ln N\ge \xi (p)+x)\) and \(\mathbb{P }(S_N/\ln N\le \xi (p)-x)\) are exactly identified, and the corresponding large deviation results follow as a consequence.
Keywords
Head-run Large deviation Hitting time Skip-free Markov chainMathematics Subject Classification (2010)
60F10 60J10Notes
Acknowledgments
The authors thank the anonymous referees for valuable comments and suggestions. Research supported in part by 985 Project, 973 Project (No. 2011CB808000), NSFC (No. 11131003), SRFDP (No. 20100003110005) and the Fundamental Research Funds for the central Universities and the Natural Science Foundation of China under grants 10971143 (Feng Wang and Xian-Yuan Wu) and 11271356 (Feng Wang and Xian-Yuan Wu).
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