Limit theorems for counting variables based on records and extremes

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, Erd?s 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.