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

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 \textit{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 far complicated. The rare results in the literature still present some flaws. This paper begins a new and complete investigation with a few number of records: (n=2\) and \(n=3\). Lessons from these simple cases will allow addressing the general formulation of simultaneous joint lower-upper records.