Asymptotic behaviour of proportions of observations in random regions determined by central order statistics from stationary processes

ABSTRACT

In this paper, we show that proportions of observations that fall into a random region determined by a given Borel set and a central order statistic converge almost surely, provided that the corresponding population quantile is unique. We also describe three types of possible asymptotic behaviour of these proportions in the case of non-unique population quantile. As an application of our findings we establish limiting properties of numbers of ties with a central order statistics in a discrete sample. Our results are derived not only for independent and identically distributed observations but more generally for strictly stationary and ergodic sequences of random variables.

KEYWORDS:

• Near order statistic observations
• central order statistics
• stationary processes
• quantiles
• almost sure convergence

AMS 2010 SUBJECT CLASSIFICATION:

• 60F15
• 60G10
• 62G30
• 62G20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

A. Dembińska was partially supported by Polish National Science Centre under Grant no. [2015/19/B/ST1/03100]; and by Warsaw University of Technology under Grant no. [504P/1120/0114/000].