[Paper Review] The largest order statistics for the inradius in an isotropic STIT tessellation

Nicolas Chenavier, Werner Nagel|arXiv (Cornell University)|Dec 27, 2018

Point processes and geometric inequalities22 references4 citations

TL;DR

This paper studies the extreme inradius values in a stationary and isotropic STIT tessellation in the plane using the Chen-Stein method for Poisson approximation. It establishes that, as the window size ρ grows, the distribution of the number of cells with inradius exceeding a threshold vρ converges in total variation to a Poisson distribution with mean τ, where vρ is chosen so that the expected number of such cells equals τ.

ABSTRACT

A planar stationary and isotropic STIT tessellation at time $t>0$ is observed in the window $W_\ ho={t^{-1}}\\sqrt{\\pi \\ \ ho}\\cdot [-\\frac{1}{2},\\frac{1}{2}]^2$, for $\ ho>0$. With each cell of the tessellation, we associate the inradius, which is the radius of the largest disk contained in the cell. Using the Chen-Stein method, we compute the limit distributions of the largest order statistics for the inradii of all cells whose nuclei are contained in $W_\ ho$ as $\ ho$ goes to infinity.

Motivation & Objective

• To analyze the asymptotic behavior of the largest inradius values in a planar stationary and isotropic STIT tessellation.
• To determine the limit distribution of the number of cells with inradius exceeding a high threshold in a growing window.
• To apply the Chen-Stein method to obtain a Poisson approximation for exceedance counts with an explicit rate of convergence.
• To extend extreme value theory to STIT tessellations by focusing on the inradius, a geometric characteristic with explicit distribution.
• To investigate the robustness of the Poisson limit under isotropy and stationarity, and to consider extensions to non-isotropic and higher-dimensional settings.

Proposed method

• Define a growing square window $ W_{ ho} = t^{-1} ho^{1/2} ho^{-1/2} [-1/2, 1/2]^2 $, scaled so that the cell intensity is $ \gamma_t = t^2 / \pi $.
• Associate each cell with its inradius $ R(z) $, the radius of the largest disk contained in the cell, and use the known result that the typical cell’s inradius is exponentially distributed with parameter $ 2t $.
• Set a threshold $ v_{ ho} = \frac{1}{2t}(\log \rho - \log \tau) $ such that the expected number of cells in $ W_{\rho} $ with inradius $ > v_{\rho} $ equals a fixed $ \tau > 0 $.
• Apply the Chen-Stein method to bound the total variation distance between the distribution of the number of exceedances $ N_{W_{\rho}}(v_{\rho}) $ and a Poisson distribution with mean $ \tau $.
• Use geometric arguments involving line processes and trapezoidal regions to estimate the dependency structure between distant cells, particularly the measure of lines separating neighboring cells.
• Establish that the error terms in the Chen-Stein bound vanish as $ \rho \to \infty $, proving convergence in total variation to the Poisson distribution.

Experimental results

Research questions

• RQ1 What is the limiting distribution of the number of cells with inradius exceeding a high threshold in a growing window of a planar STIT tessellation?
• RQ2 How does the Chen-Stein method enable a precise rate of convergence for Poisson approximation in this spatial extremal setting?
• RQ3 To what extent does isotropy and stationarity affect the asymptotic behavior of extreme inradius statistics in STIT tessellations?
• RQ4 Can the Poisson approximation for exceedance counts be extended to non-isotropic STIT tessellations or higher dimensions?
• RQ5 How does the choice of threshold $ v_{\rho} $, dependent on $ \rho $, ensure a non-degenerate limit for the number of exceedances?

Key findings

• The number of cells in the window $ W_{\rho} $ with inradius exceeding $ v_{\rho} = \frac{1}{2t}(\log \rho - \log \tau) $ converges in total variation to a Poisson distribution with mean $ \tau $ as $ \rho \to \infty $.
• The convergence rate of the Poisson approximation is explicitly bounded via the Chen-Stein method, which provides a quantitative error estimate.
• The inradius of the typical cell in a STIT tessellation is exponentially distributed with parameter $ 2t $, a key input for the threshold choice.
• The proof relies on controlling the dependency between distant cells using geometric measures of separating lines, particularly through the use of trapezoidal regions.
• The total variation distance between the distribution of exceedance counts and the Poisson limit tends to zero as $ \rho \to \infty $, confirming the asymptotic Poisson behavior.
• The result remains valid for general convex windows with non-empty interior, not just square-shaped ones, and extends to non-isotropic STIT tessellations under mild directional conditions.