arXiv subjects

Justin Baars

Publications and source records attributed to Justin Baars.

3 recordsLinked to original sources

arXiv2025

Asymptotically distribution-free goodness-of-fit testing for point processes

Consider an observation of a multivariate temporal point process $N$ with law $\mathcal P$ on the time interval $[0,T]$. To test the null hypothesis that $\mathcal P$ belongs to a given parametric family, we construct a convergent compensated counting process to which we apply an innovation martingale transformation. We prove that the resulting process converges weakly to a standard Wiener process. Consequently, taking a suitable functional of this process yields an asymptotically distribution-free goodness-of-fit test for point processes. For several standard tests based on the increments of this transformed process, we establish consistency under alternative hypotheses. Finally, we assess the performance of the proposed testing procedure through a Monte Carlo simulation study and illustrate its practical utility with two real-data examples.

math.ST↗

arXiv2024

Spatiotemporal Hawkes processes with a graphon-induced connectivity structure

We introduce a spatiotemporal self-exciting point process $(N_t(x))$, boundedly finite both over time $[0,\infty)$ and space $\mathscr X$, with excitation structure determined by a graphon $W$ on $\mathscr{X}^2$. This graphon Hawkes process generalizes both the multivariate Hawkes process and the Hawkes process on a countable network, and despite being infinite-dimensional, it is surprisingly tractable. After proving existence, uniqueness and stability results, we show, both in the annealed and in the quenched case, that for compact, Euclidean $\mathscr X\subset\mathbb R^m$, any graphon Hawkes process can be obtained as the suitable limit of $d$-dimensional Hawkes processes $\tilde N^d$, as $d\to\infty$. Furthermore, in the stable regime, we establish an FLLN and an FCLT for our infinite-dimensional process on compact $\mathscr X\subset\mathbb R^m$, while in the unstable regime we prove divergence of $N_T(\mathscr X)/T$, as $T\to\infty$. Finally, we exploit a cluster representation to derive fixed-point equations for the Laplace functional of $N$, for which we set up a recursive approximation procedure. We apply these results to show that, starting with multivariate Hawkes processes $\tilde N^d_t$ converging to stable graphon Hawkes processes, the limits $d\to\infty$ and $t\to\infty$ commute.

math.PR↗

arXiv2023

Delayed Hawkes birth-death processes

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.

math.PR↗