Hawkes autoregressive processes: a new model for multiscale and heterogeneous processes

TL;DR

This work introduces Hawkes AutoRegressive (HAR) processes that couple a continuous-time Hawkes intensity with a multiscale autoregressive component to model heterogeneous data across timescales. It provides a rigorous probabilistic foundation, proving existence of non-exploding and stationary HAR processes under spectral-radius conditions, and develops a linear cluster representation with exponential moment bounds. The authors extend LASSO-based estimation to HAR by using adaptive weights and a dictionary of bounded functions, establishing Oracle inequalities under Restricted Eigenvalue conditions and deriving data-driven calibration via Bernstein inequalities. A Berbee coupling framework yields exponential concentration results and supports sharp convergence rates to stationarity, enabling robust statistical guarantees for inference in multiscale heterogeneous point-process models. The results have potential applications in neuroscience and other domains where coupled continuous-time and discrete-time dynamics arise, and they offer a principled way to perform sparse interaction selection via LASSO in HAR settings that include cross-scale dependencies.

Abstract

Both Hawkes processes and autoregressive processes depend on linear functionals of their past while modelling different types of data. As different datasets obtained through the recording of the same phenomena may be heterogeneous and occur at different timescales, it is important to study multiscale and heterogenous processes, such as those obtained by combining Hawkes and autoregressive processes. In this paper, we present probabilistic results for this new Hawkes autoregressive (HAR) model, including the existence of a stationary version, a cluster representation, exponential moments and asymptotic behaviour. We also derive statistical results for estimating interactions, extending the well-known LASSO estimation method to Hawkes Autoregressive (HAR) processes.

Figures (4)

• Figure 1: Illustration of the clusters. The red dots are the immigrant points. They arrive with the immigrant rate and give birth to clusters, represented by the blue shapes.
• Figure 2: Illustration of a cluster. The tree $\mathcal{T}$ is represented with the types and birth dates. The ancestor is $u=\varnothing$, it has two children, $u=0$ and $u=1$. $u=0$ has one child ($u=00$), $u=1$ has three children ($u=10$, $u=11$ and $u=12$) and so on.
• Figure 3: Schematic representation of the coupling $\mathbb{X}^{(n,\ell,\phi)}$. The dotted part in red or green represent the unused part of the $\bm{{\mathop{\mathrm{HAR}}\nolimits}}_{\mathfrak{M}}\bm{(} jn+\phi-\ell \bm{)}$ processes. Only the solid lines are part of the coupling. The blue lines represent the independent random variables associated to $j-2$ and $j$ of Proposition \ref{['prop6.6']}.
• Figure 4: This figure illustrates node $m$. In red there is $\lambda^m_t$, dotted in red there is the temporary intensity $\mathop{\mathrm{Temp}}\nolimits_{t_n}(\lambda^m)_t$. Green dots are the pre-points of node $m$ belonging to the set $E_0$. Step 2 is re-run for the first time at time $s_1$ : time appearance of the first pre-point at level $s_0$ (ie in $E_0$). Dotted in blue there is the temporary intensity at level $s_1$ calculated in Step 2. The shaded area is where there might be some new pre-points (pre-points at level $s_1$ that weren't pre-points at level $s_0=t_n$), here there is one represented by a blue dot.

Theorems & Definitions (100)

• Definition 2.1
• Definition 3.1: Non exploding process
• Definition 3.2: Integrable initial condition
• Definition 3.3: Modification
• Theorem 3.4: Existence of non exploding HAR processes
• Remark 3.5
• Definition 3.6: 1-stationary
• Remark 3.7
• Definition 3.8
• Theorem 3.9: Stationary HAR processes