Delayed Hawkes birth-death processes
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911070878171136 |
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| author | Baars, Justin Laeven, Roger J. A. Mandjes, Michel |
| author_facet | Baars, Justin Laeven, Roger J. A. Mandjes, Michel |
| contents | We introduce, and formally establish, a variant of the Hawkes-fed birth-death process -- the delayed Hawkes birth-death process -- in which the conditional intensity does not increase at arrivals but at departures from the system. In a scaling limit where sojourn times are stretched out by a factor $\sqrt T$, after which time gets contracted by a factor $T$, the delayed Hawkes process behaves markedly differently from its classical counterpart. We design a family of models admitting a cluster representation and containing the Hawkes and delayed Hawkes processes as special cases. The cluster representation allows for transform characterizations by a fixed-point equation and for analysis of heavy-tailed asymptotics. We compare the delayed Hawkes process to the classical Hawkes process using stochastic ordering, which enables us to describe stationary distributions and heavy-traffic behavior. In the Markovian network case, a recursive procedure is presented to calculate the $d$th-order moments analytically. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12812 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Delayed Hawkes birth-death processes Baars, Justin Laeven, Roger J. A. Mandjes, Michel Probability 60G55, 60E10, 60E15, 62E20 We introduce, and formally establish, a variant of the Hawkes-fed birth-death process -- the delayed Hawkes birth-death process -- in which the conditional intensity does not increase at arrivals but at departures from the system. In a scaling limit where sojourn times are stretched out by a factor $\sqrt T$, after which time gets contracted by a factor $T$, the delayed Hawkes process behaves markedly differently from its classical counterpart. We design a family of models admitting a cluster representation and containing the Hawkes and delayed Hawkes processes as special cases. The cluster representation allows for transform characterizations by a fixed-point equation and for analysis of heavy-tailed asymptotics. We compare the delayed Hawkes process to the classical Hawkes process using stochastic ordering, which enables us to describe stationary distributions and heavy-traffic behavior. In the Markovian network case, a recursive procedure is presented to calculate the $d$th-order moments analytically. |
| title | Delayed Hawkes birth-death processes |
| topic | Probability 60G55, 60E10, 60E15, 62E20 |
| url | https://arxiv.org/abs/2306.12812 |