We present a relatively simple and mostly elementary proof of the Lévy–Khintchine formula for subordinators. The main idea is to study the Poisson process time-changed by the subordinator. This is a compound Poisson process which is easy to investigate using elementary probabilistic techniques. It turns out that its rate equals the value of the Laplace exponent of the leading subordinator at 1, and all other characteristics of the subordinator affect just the distribution of summands. The technical tools used are conditional expectations, probability generating function and convergence of discrete random variables.
Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Compound Poisson process
- Lévy–Khintchine formula