Convergence rates for record times and the associated counting process
Let \(X,X_ 1,X_ 2,..\). be i.i.d. random variables with a continuous distribution function F. The record times L(n) are defined recursively by: \[ L(1)=1\text{ and } L(n)=\min \{k:\;X_ k>X_{L(n-1)}\},\quad n\geq 2. \] The counting process \(\{\mu\) (n), \(n\geq 1\}\), which is defined by the equality \(\mu (n)=\max \{k:\) L(k)\(\leq n\}\), is studied. Namely, the rate of convergence in limit relations for \(\mu\) (n) is given. As an example, the following estimate is presented: \[ \sup_{k} | P\{L(n)>k\}-\phi ((n-\log k)/\sqrt{n})| \leq 43/\sqrt{n}, \] where \(\phi\) is the d.f. of a standard normal random variable.
scientific article; zbMATH DE number 4166177
scientific article