Strong approximation of the number of renewal paced record times
Let \(Y_ 0, Y_ 1, Y_ 2, \dots\) and positive \(Z_ 1, Z_ 2 \dots\) be independent random variables, the \(Y_ i\) with common continuous d.f. and \(Z_ 2, Z_ 3, \dots\) i.i.d. with \(E \exp (tZ_ 2) < \infty\) in a \(t\)-neighbourhood of zero. Put \(S_ 0 = 0\), \(S_ n = Z_ 1 + \cdots + Z_ n\), \(n \geq 1\), and \(N(t) = \# \{n \geq 0 \mid S_ n \leq t\}\). It is assumed that \(Y_ i\) is observed at time \(S_ i\), \(i = 0,1,2, \dots\) The paper studies the process of record times of these observations. They form a thinning of the renewal process \(N(t)\), the observation at time \(S_ n\) being a record with probability \(1/(n + 1)\) independent of everything else. By definition \(Y_ 0\) at time \(S_ 0 = 0\) is a record. Let \(H(t)\) be the number of records observed in \([0,t]\) and let \(a(n) = \sum^ n_{i = 0} (n + 1)^{-1}\) and \(b(n) = \sum^ n_{i = 0} n(n + 1)^{-2}\) be the expectation and variance of the number of records in \(Y_ 0, \dots, Y_ n\). Then \(A_ 1(t) = a(N(t)) = E \{H(t) \mid N(t)\}\) and \(B_ 1(t) = b(N(t)) = \text{Var} \{H(t) \mid N(t)\}\). Theorems: A central limit theorem for \((B_ 1(t))^{-1/2} (H(t) - A_ 1 (t))\) with non-constant convergence bound. Then a strong approximation theorem for \(H(t) - A_ 1 (t)\) by \(W(B_ 1 (t))\) where \(W(t)\) is a standard Wiener process, and a strong approximation theorem for \(H(t) - \log t\) by \(W (\log t)\). Corollaries are loglog laws and a.s. order estimates in the strong approximations. There is a large list of references on records and strong approximation.
scientific article; zbMATH DE number 758075
scientific article