{
  "id": "1706.03975",
  "version": "v1",
  "published": "2017-06-13T09:45:59.000Z",
  "updated": "2017-06-13T09:45:59.000Z",
  "title": "A note on critical Hawkes processes",
  "authors": [
    "Matthias Kirchner"
  ],
  "comment": "8 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-13T09:45:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distribution function",
      "multitype critical hawkes processes",
      "construction",
      "cluster-invariant point processes",
      "independent cluster operations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}