Infinite Size-Biased Orders

TL;DR

This paper introduces an infinite size-biased order using exponential variables, explores its properties, and examines how its order type varies with parameters, contributing to the understanding of infinite random orders.

Contribution

It presents a new construction of infinite size-biased orders with arbitrary parameters and analyzes their properties and order types.

Findings

The order type depends on the size parameters.

Basic properties and constructions are established.

The order type can be ${ m Z}_{>0}$, ${ m Q}$, or others.

Abstract

The infinite random size-biased order with arbitrary positive size parameters is introduced in terms of independent exponential random variables. We collect basic properties and constructions of the order, some of which belong to the folklore, and show how the order type (e.g. ${\mathbb Z}_{>0}, {\mathbb Q}$ or any other possible) depends on parameters.