# Variational series

series of order statistics

An arrangement of the values of a random sample $( x _ {1} \dots x _ {n} )$ with distribution function $F( x)$ in ascending sequence $x _ {(} 1) \leq \dots \leq x _ {(} n)$. The series is used to construct the empirical distribution function ${F _ {n} } ( x) = {m _ {x} } /n$, where $m _ {x}$ is the number of terms of the series which are smaller than $x$. Important characteristics of series of order statistics are its extremal terms ( $x _ {(} 1) = \min _ {1 \leq i \leq n } x _ {i}$, $x _ {(} n) = \max _ {1 \leq i \leq n } x _ {i}$) and the range $R _ {n} = {x _ {(} n) } - {x _ {(} 1) }$. The densities of the distributions of the minimum and maximum terms of a series of order statistics in the case

$$F ( x) = \int\limits _ {- \infty } ^ { x } p ( y) dy$$

are defined by the expressions

$$p _ {(} 1) ( x) = n [ 1 - F ( x)] ^ {n - 1 } p ( x)$$

and

$$p _ {(} n) ( x) = nF ^ { n - 1 } ( x) p( x).$$

Considered as a stochastic process with time index $i$, $i = 1 \dots n$, the series of order statistics forms a non-homogeneous Markov chain.