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A simple proof of the Lévy–Khintchine formula for subordinators

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  • Yakubovich, Yuri

Abstract

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.

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  • Yakubovich, Yuri, 2021. "A simple proof of the Lévy–Khintchine formula for subordinators," Statistics & Probability Letters, Elsevier, vol. 176(C).
    • Handle: RePEc:eee:stapro:v:176:y:2021:i:c:s0167715221000985
    DOI: 10.1016/j.spl.2021.109136
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    References listed on IDEAS

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    1. L. Beghin & P. Vellaisamy, 2018. "Space-Fractional Versions of the Negative Binomial and Polya-Type Processes," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 463-485, June.
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