2017/06/13 by Matthias Kirchner, Kirchner, Matthias
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Diffusion and Search Dynamics #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1706.03975
openalex publication_date 2017/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F be a distribution function on ℝ with F(0) = 0 and density f. Let F be the distribution function of X1 - X2, Xi∼ F, i=1,2, iid. We show that for a critical Hawkes process with displacement density (= `excitement function' = `decay kernel') f, the random walk induced by F is necessarily transient. Our conjecture is that this condition is also sufficient for existence of a critical Hawkes process. Our train of thought relies on the interpretation of critical Hawkes processes as cluster-invariant point processes. From this property, we identify the law of critical Hawkes processes as a limit of independent cluster operations. We establish uniqueness, stationarity, and infinite divisibility. Furthermore, we provide various constructions: a Poisson embedding, a representation as Hawkes process with renewal immigration, and a backward construction yielding a Palm version of the critical Hawkes process. We give specific examples of the constructions, where F is regularly varying with tail index α∈(0,0.5). Finally, we propose to encode the genealogical structure of a critical Hawkes process with Kesten (size-biased) trees. The presented methods lay the grounds for the open discussion of multitype critical Hawkes processes as well as of critical integer-valued autoregressive time series.