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Estimating non-linear functionals of trawl processes

Orimar Sauri

Abstract

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.

Estimating non-linear functionals of trawl processes

Abstract

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 and , where represents the trawl function of interest and a non-linear test function. We show that our estimator for is consistent and asymptotically Gaussian regardless of the memory of the process. We further demonstrate that the same phenomenon occurs for the estimation of as long as , as , for some . Additionally, we illustrate how our results can be used to construct a test statistic robust to memory effects for the presence of -dependent.

Paper Structure

This paper contains 10 sections, 16 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Estimating non-linear functionals of trawl functions
  4. Limit theorems for $\Psi^{n}(g)$ and $\Lambda^{n}(g)$
  5. A test for $T$-dependent trawls
  6. Proofs
  7. Preliminary estimates and decompositions
  8. Proof of Theorems \ref{['consitency_Phin']} - \ref{['consitency_Lambdan_bar']}
  9. Proof of Theorems \ref{['CLT_Phin']} and \ref{['CLT_Lambdan_hatLambdan']}
  10. Approximations of the Asymptotic Variance

Key Result

Theorem 1

Let $g:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function of polynomial growth of order $q\geq0$. Suppose that $\mathbb{E}(\rvert L^{\prime}\rvert^{2q\lor4})<\infty$. Then, $\Psi^{n}(g)\overset{u.c.p}{\rightarrow}\Psi(g).$

Theorems & Definitions (31)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Remark 2
  • Proposition 1
  • proof
  • ...and 21 more