Estimating non-linear functionals of trawl processes

Trawl processes are a family of continuous-time, infinitely divisible, stationary processes whose correlation structure is entirely characterized by their so-called trawl function. This paper investigates the problem of estimating non-linear functionals of a trawl function under in-fill and long-span sampling schemes. Specifically, building on the work of \cite{SauriVeraart23}, we introduce non-parametric estimators for functionals of the type $Ψ_{t}(g)=\int_{0}^{t}g(a(s))\mathrm{d}s$ and $ Λ_t(g)=\int_{t}^{\infty}g(a(s))\mathrm{d}s$, where $a$ represents the trawl function of interest and $g$ a non-linear test function. We show that our estimator for $Ψ_{t}(g)$ is consistent and asymptotically Gaussian regardless of the memory of the process. We further demonstrate that the same phenomenon occurs for the estimation of $Λ_t(g)$ as long as $g(x)= \mathrm{O} (\lvert x\rvert^p)$, as $x\to0$, for some $p>3$. Additionally, we illustrate how our results can be used to construct a test statistic robust to memory effects for the presence of $T$-dependent.

💡 Research Summary

This paper addresses the non‑parametric estimation of two nonlinear functionals of the trawl function a in a continuous‑time, infinitely divisible, stationary trawl process Xₜ = L(Aₜ). The functionals of interest are

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