Hawkes Identification with a Prescribed Causal Basis: Closed-Form Estimators and Asymptotics

Researchers Xinhui Rong and Girish N. Nair have published a groundbreaking paper introducing a closed-form Least Squares identification framework for Hawkes processes with prescribed basis kernels. This addresses a critical bottleneck in analyzing complex event sequences with memory—common in neural activity modeling, financial transactions, and social media interactions. Traditional Hawkes process identification suffers from nonlinear optimization challenges, often requiring iterative likelihood methods that lack theoretical guarantees under model misspecification. The new framework transforms this into a linear parameterization problem while maintaining rigorous statistical foundations.

The technical breakthrough lies in proving the almost-sure positive definiteness of the empirical Gram matrix, which guarantees estimator existence. The researchers provide complete asymptotic theory: convergence to true parameters under correct specification, convergence to well-defined pseudo-true parameters under misspecification, and explicit Central Limit Theorems for both regimes. This enables practitioners to analyze confidence intervals and statistical significance directly. The method's computational efficiency—bypassing iterative optimization—makes it particularly valuable for real-time applications in algorithmic trading, network monitoring, and brain-computer interfaces where traditional methods are too slow.