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Local-asymptotic limit law with unknown diffusivity levels under vanishing jump

Mathematical StatisticsProbability TheoryLearning Theory

Sourced from the work of Markus Reiß, Claudia Strauch, Lukas Trottner

§ Problem Statement

Setup

Let (Ω,F,(Ft)t∈[0,T],P)(\Omega,\mathcal F,(\mathcal F_t)_{t\in[0,T]},\mathbb P) carry a cylindrical Wiener process WW on L2(0,1)L^2(0,1), fix T>0T>0, and consider the stochastic heat equation in weighted-Laplacian form

dXt=ΔϑXt dt+dWt,Δϑf:=∂x(ϑ(x)∂xf),dX_t=\Delta_{\vartheta}X_t\,dt+dW_t,\qquad \Delta_{\vartheta}f:=\partial_x\big(\vartheta(x)\partial_x f\big),

on (0,1)(0,1) with Dirichlet boundary conditions and deterministic initial condition. Assume a one-change parametrization

ϑ(x)=ϑ∘+h 1[τ,1](x),τ∈(ε,1−ε), ϑ∘∈[ϑ‾,ϑ‾], ϑ∘+h∈[ϑ‾,ϑ‾].\vartheta(x)=\vartheta^{\circ}+h\,\mathbf 1_{[\tau,1]}(x),\qquad \tau\in(\varepsilon,1-\varepsilon),\ \vartheta^{\circ}\in[\underline\vartheta,\overline\vartheta],\ \vartheta^{\circ}+h\in[\underline\vartheta,\overline\vartheta].

This setup follows Reiß et al. (2023).

For localization scale δ↓0\delta\downarrow0, let Kδ,x0K_{\delta,x_0} be the local measurement kernel(s) used in the source model and observe continuously in time the localized process

Xδ,x0(t):=⟨Xt,Kδ,x0⟩,X_{\delta,x_0}(t):=\langle X_t,K_{\delta,x_0}\rangle,

together with the associated localized Laplacian term

Xδ,x0Δ(t):=⟨Xt,ΔKδ,x0⟩X^{\Delta}_{\delta,x_0}(t):=\langle X_t,\Delta K_{\delta,x_0}\rangle

(and analogously for the finite collection of observation points/kernels in the experiment).

In the vanishing-jump regime h=hδ→0h=h_\delta\to0, under contiguous scaling where an oracle limit law is available when nuisance parameters are fixed, define the simultaneous M-estimator

(τ^δ,h^δ,ϑ^δ∘)(\hat\tau_\delta,\hat h_\delta,\hat\vartheta^{\circ}_\delta)

as any measurable maximizer of the source contrast (equivalently Gaussian quasi-likelihood) built from the local kernel observations and their XΔX^{\Delta} regressors.

Unsolved Problem

Determine deterministic normalizations and a nondegenerate explicit joint weak limit law for the centered/scaled vector

(τ^δ−τ, h^δ−hδ, ϑ^δ∘−ϑ∘)\big(\hat\tau_\delta-\tau,\ \hat h_\delta-h_\delta,\ \hat\vartheta^{\circ}_\delta-\vartheta^{\circ}\big)

in this contiguous vanishing-jump regime, and quantify the efficiency loss (if any) of the change-point coordinate relative to the oracle benchmark with nuisance parameters known.

§ Discussion

Loading discussion…

§ Significance & Implications

The discussion in Reiß et al. (2023) establishes the faint-signal limit theorem under a simplifying known-diffusivity setup; the simultaneous unknown-nuisance regime is therefore an important remaining inferential regime near detectability. Resolving it would quantify the cost of nuisance estimation at change-point scale and sharpen the local asymptotic theory for this observation model.

§ Known Partial Results

  • Reiß et al. (2023): The source paper derives consistency and rates for simultaneous M-estimation (including a nuisance baseline diffusivity parameter) and proves a vanishing-jump limit theorem in a simplified oracle setting with known diffusivity components.

§ References

[1]

Change Point Estimation for a Stochastic Heat Equation

Markus Reiß, Claudia Strauch, Lukas Trottner (2023)

arXiv preprint (journal publication in Annals of Statistics reported separately; publication year not encoded here)

📍 arXiv:2307.10960v2, Section 4 (Discussion), Perspectives paragraph on the vanishing-jump regime with known diffusivity constants; PDF page 21 (printed page 22 in manuscript pagination).

Primary source for the model, estimators, and discussion of the vanishing-jump regime.

Link ↗arXiv ↗

§ Tags

spdechange-pointm-estimationlocal-asymptoticsstochastic-heat-equation
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