# Relaxation of monotone coupling conditions: Poisson approximation and beyond

## TL;DR

This paper relaxes the monotonicity assumptions in size-bias couplings to establish explicit Poisson approximation bounds for various dependent random variables, broadening applicability to noisy models and other approximations.

## Contribution

It introduces methods to relax monotonicity conditions in size-bias couplings, enabling Poisson approximation bounds based on first two moments for more general dependent variables.

## Key findings

- Effective bounds for models with noise contamination
- Poisson approximation for associated or negatively associated variables
- Extensions to Poincaré inequality and normal approximation

## Abstract

It is well-known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also give explicit Poisson approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincar\'e inequality and in a normal approximation result.