Simultaneous Joint Lower and Upper record values Probability Laws for Absolutely Continuous or Discrete Data

TL;DR

This paper studies the probability laws of joint lower and upper record values in sequences of independent random variables, addressing gaps in the literature by analyzing cases with two and three records to develop a comprehensive understanding.

Contribution

It provides a new, complete investigation into the joint distribution of lower and upper record values, focusing on simple cases to facilitate generalization.

Findings

Derived explicit probability density functions for joint lower-upper records with 2 and 3 records.

Identified key challenges and gaps in existing literature on joint record distributions.

Laid groundwork for general formulation of simultaneous joint lower-upper record laws.

Abstract

This paper investigates the probability density function ($pdf$) of the $(2n-1)$-vector $(n\geq 1)$ of both lower and upper record values for a sequence of independent random variables with common $pdf f$ defined on the same probability space, provided that the lower and upper record times are finite up to $n$. A lot is known about the lower or the upper record values when they are studied separately. When put together, the challenges are a far bigger complicated. The rare results in the literature still present important flaws. This paper begins a new and complete investigation with a few number of records: 2 and 3. Lessons from these simple cases will allow addressing the general formulation of simultaneous joint lower-upper records.