# On a limit theorem for a non-linear scaling

## TL;DR

This paper establishes a limit theorem for sums of i.i.d. positive random variables scaled by a non-linear transform, showing the limits are either degenerate or compound Poisson distributions.

## Contribution

It introduces a novel non-linear scaling method and characterizes the possible weak limit distributions for sums of i.i.d. variables under this transformation.

## Key findings

- Weak limits are either degenerate or compound Poisson distributions.
- The non-linear transform used is $ ext{max}(0, x - r)$.
- The result extends classical limit theorems to non-linear scaling contexts.

## Abstract

In this note, we proved that weak limits, of sums of independent positive identically distributed random variables which are re-normalized by a non-linear shrinking transform $\max(0, x-r)$, are either degenerate or (some) compound Poisson distributions.