Convergence rates for record times and the associated counting process
type: Item
instance of: scholarly article
title: Convergence rates for record times and the associated counting process
zbMATH Open document ID: 0709.60018
DOI: 10.1016/0304-4149(90)90047-V
author: Allan Gut
published in: Stochastic Processes and their Applications
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G99
Mathematics Subject Classification ID: 60F25
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 4166177
zbMATH Keywords: counting process
zbMATH Keywords: central limit theorem
zbMATH Keywords: record times
zbMATH Keywords: rate of convergence
reviewed by: V. Paulauskas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4166177