# A simple characterization of tightness for convex solid sets of positive random variables

## TL;DR

This paper characterizes tightness of convex solid sets of positive random variables through radial boundedness, providing a necessary and sufficient condition relevant to mathematical finance applications.

## Contribution

It introduces a simple, precise criterion linking tightness and radial boundedness for convex solid sets of positive random variables.

## Key findings

- Tightness is equivalent to radial boundedness in convex solid sets.
- Radial boundedness ensures boundedness in probability for these sets.
- The result has implications for mathematical finance models.

## Abstract

We show that for a convex solid set of positive random variables to be tight, or equivalently bounded in probability, it is necessary and sufficient that it is radially bounded, i.e. that every ray passing through one of its elements eventually leaves the set. The result is motivated by problems arising in mathematical finance.