A note on critical Hawkes processes

Let $F$ be a distribution function on $\mathbb{R}$ with $F(0) = 0$ and density $f$. Let $\tilde{F}$ be the distribution function of $X_1 - X_2$, $X_i\sim F,\, i=1,2,\text{ iid}$. We show that for a critical Hawkes process with displacement density (= `excitement function' = `decay kernel') $f$, the random walk induced by $\tilde{F}$ 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 $\alpha\in(0,0.5)$. 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.