A simple proof of the L\'evy--Khintchine formula for subordinators

TL;DR

This paper provides an accessible and elementary proof of the Lévý--Khintchine formula specifically for subordinators, utilizing Poisson process time-changes and basic probabilistic tools.

Contribution

It introduces a simplified proof of the Lévý--Khintchine formula for subordinators, making the result more accessible and easier to understand.

Findings

Proof is elementary and accessible

Uses Poisson process time-change technique

Employs basic probabilistic tools

Abstract

We present a relatively simple and mostly elementary proof of the L\'evy--Khintchine formula for subordinators. The main idea is to study the Poisson process time-changed by the subordinator. The technical tools used are conditional expectations, probability generating function and convergence of discrete random variables.