Asymptotic behaviour of a number of repeated records
Let \(\{X_i\}_{i\in\mathbb{N}}\) be a sequence of i.i.d. random variables with continuous distribution function \(F\), let \(M_n= \max\{X_i\mid i\leq n\}\) be the record value at time \(n\) and \(\tau_n=\inf\{i\mid X_i= M_n\}\) be the moment when this record is obtained. For each \(n\in\mathbb{N}\) and \(\varepsilon>0\), \(Z_n:= \sum_{\tau_n\leq i\leq n}1_{\{M_n-\varepsilon< X_i\leq M_n\}}\) counts the number of repetitions of the record with \(\varepsilon\) accuracy since the time it was obtained and \(S_n:= \sum_{\tau_n\leq i\leq n}1_{\{M_{\tau_n-1}< X_i\leq M_{\tau_n}\}}\) counts the number of trials after \(\tau_n\) when the previous record was exceeded. The exact and asymptotic distributions of \(Z_n\) and \(S_n\) are calculated. To be more precise it is shown that \[ P(\{Z_n= k\})= {1\over n} \int^\infty_{-\infty} \sum^{n-1}_{l= k} B\Biggl(l,{F(x)- F(x-\varepsilon)\over F(x)}; h-1\Biggr)dF^n(x) \] with \(B(m,p;j):= {m\choose j}p^j(1- p)^{m- j}\) and \[ P(\{S_n= k\})= {1\over k(k+1)}+ \delta_{k1}\cdot{1\over n}. \] Thus the limiting distribution of \(S_n\) is the so-called Zipf's law. Moreover, following \textit{J. J. A. M. Brands}, \textit{F. W. Steutel} and \textit{R. J. G. Wilms} [ibid. 20, No. 3, 209-217 (1994; Zbl 0802.60048)], the a.s. convergence of \(Z_n\) is discussed.
scientific article; zbMATH DE number 1054097
scientific article