# Almost sure asymptotic properties of central order statistics from stationary processes

## TL;DR

This paper investigates the almost sure asymptotic behavior of central order statistics derived from stationary processes, introducing new properties of conditional quantiles and a strong ergodic theorem for these statistics.

## Contribution

It formulates new properties of conditional quantiles and establishes a novel strong ergodic theorem for central order statistics in stationary processes.

## Key findings

- New properties of conditional quantiles are established.
- A new version of the strong ergodic theorem for central order statistics is proved.
- Results enhance understanding of asymptotic behavior in stationary processes.

## Abstract

In this paper, we formulate and prove new properties of conditional quantiles given one of the particular sigma-fields. Next, we use them to investigate almost sure asymptotic behavior of central order statistics which arise from strictly stationary processes. Specifically we provide a new version of a strong ergodic theorem for central order statistics.