The Annals of Applied Probability

Combinatorial Lévy processes

Harry Crane

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Combinatorial Lévy processes evolve on general state spaces of combinatorial structures, of which standard examples include processes on sets, graphs and $n$-ary relations and more general possibilities are given by processes on graphs with community structure and multilayer networks. In this setting, the usual Lévy process properties of stationary, independent increments are defined in an unconventional way in terms of the symmetric difference operation on sets. The main theorems characterize both finite and infinite state space combinatorial Lévy processes by a unique $\sigma$-finite measure. Under the additional assumption of exchangeability, I prove a more explicit characterization by which every exchangeable combinatorial Lévy process corresponds to a Poisson point process on the same state space. Associated behavior of the projection into a space of limiting objects reflects certain structural features of the covering process.

Article information

Ann. Appl. Probab., Volume 28, Number 1 (2018), 285-339.

Received: January 2015
Revised: November 2016
First available in Project Euclid: 3 March 2018

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Zentralblatt MATH identifier

Primary: 60B05: Probability measures on topological spaces 60G09: Exchangeability 60G51: Processes with independent increments; Lévy processes 60J05: Discrete-time Markov processes on general state spaces 60J25: Continuous-time Markov processes on general state spaces

Combinatorial stochastic process Lévy process dynamic networks Lévy–Itô–Khintchine representation exchangeability finite exchangeability


Crane, Harry. Combinatorial Lévy processes. Ann. Appl. Probab. 28 (2018), no. 1, 285--339. doi:10.1214/17-AAP1306.

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