# Some Results on the Additivity and Multiplication Order Preserving Properties of Stochastic Orders

## TL;DR

This paper investigates how stochastic orders behave under addition and multiplication, proving that the usual stochastic order is preserved in these operations and exploring similar properties for other stochastic orders.

## Contribution

It establishes the additivity and multiplicativity of the usual stochastic order and examines these properties for nine other univariate stochastic orders, providing counterexamples.

## Key findings

- Usual stochastic order is additive and multiplicative.
- Counterexamples for nine other stochastic orders.
- Open problems remain for some stochastic orders.

## Abstract

In this paper, we consider the problem of order preservation under addition and multiplication operators over the vector space of univariate real-valued random variables. Consistent with the case of usual order over the real numbers-as constant random variables, we prove that the usual stochastic order is both additive and multiplicative. We additionally discuss the situation for nine extra univariate stochastic orders presenting counterexamples in some cases. The problem remains unsolved for other univariate stochastic orders.