Limit theorems for counting variables based on records and extremes

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• 作者：Allan Gut ; Ulrich Stadtmüller
• 关键词：Record times ; Records ; Extremes ; Counting process ; Weak convergence
• 刊名：Extremes
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：20
• 期：1
• 页码：33-52
• 全文大小：313KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Statistics, general; Quality Control, Reliability, Safety and Risk; Civil Engineering; Hydrogeology; Environmental Management; Statistics for Business/Economics/Mathematical Finance/Insurance;
• 出版者：Springer US
• ISSN：1572-915X
• 卷排序：20

文摘

Hsu and Robbins (Proc. Nat. Acad. Sci. USA 33, 25–31, 1947) introduced the concept of complete convergence as a complement to the Kolmogorov strong law, in that they proved that \( {\sum }_{n=1}^{\infty } P(|S_{n}|>n\varepsilon )<\infty \) provided the mean of the summands is zero and that the variance is finite. Later, Erdős proved the necessity. Heyde (J. Appl. Probab. 12, 173–175, 1975) proved that, under the same conditions, \(\lim _{\varepsilon \searrow 0} \varepsilon ^{2}{\sum }_{n=1}^{\infty } P(| S_{n}| \geq n\varepsilon )=EX^{2}\), thereby opening an area of research which has been called precise asymptotics. Both results above have been extended and generalized in various directions. Some time ago, Kao proved a pointwise version of Heyde’s result, viz., for the counting process \(N(\varepsilon ) ={\sum }_{n=1}^{\infty }1\hspace *{-1.0mm}\text {{I}} \{|S_{n}|>n\varepsilon \}\), he showed that \(\lim _{\varepsilon \searrow 0} \varepsilon ^{2} N(\varepsilon )\overset {d}{\to } E\,X^{2}{\int }_{0}^{\infty } 1\hspace *{-1.0mm}\text {I}\{|W(u)|>u\}\,du\), where W(⋅) is the standard Wiener process. In this paper we prove analogs for extremes and records for i.i.d. random variables with a continuous distribution function.