Upper bounds for spatial point process approximations. (English) Zbl 1067.60022

Schuhmacher, Dominic

Ann. Appl. Probab. 15, No. 1B, 615-651 (2005).

Let \(D_1\), \(D_2\in \mathbb N=\{1,2,3,...\}\) and \(D=D_1 + D_2\). Consider a point process \(\xi\) on \(\mathbb R^D=\mathbb R^{D_1}\times \mathbb R^{D_2}\), which has expectation measure \(\nu\) and meets three conditions, namely, 1) absolute continuity of \(\nu\) with a mild restriction on the density; 2) an orderliness condition in the \(\mathbb R^{D_1}\)-directions [for a detailed account of orderliness, see D. J. Daley, in: Stochastic Geometry, Tribute Memory Rollo Davidson, 148-161 (1974; Zbl 0285.60039)]; 3) a mixing condition in the \(\mathbb R^{D_2}\)- directions. The various versions of the third condition are mixing conditions of different strength [see P. Doukhan, “Mixing: Properties and examples” (1994; Zbl 0801.60027)].
Let \(\eta\) be a Poisson process with the same expectation measure and let \(\theta_T: \mathbb R^D\to \mathbb R^D\) be the linear transformation that stretches the first \(D_1\) coordinates by a factor \(\omega(T)^{1/D_1}\) and compresses the last \(D_2\) coordinates by a factor \(T^{1/D_2}\). Consider restrictions of transformed processes \(\xi \theta_T^{-1}\) and \(\eta \theta_T^{-1}\) to a bounded cube \(J=[-1, 1)^D\). It was shown by S. P. Ellis [Adv. Appl. Probab. 18, 646–659 (1986; Zbl 0609.60059)] that, for bounded measurable functions \(f_T: J\to \mathbb R\) with \(\| f_T\| = O(\sqrt{\omega(T)/T})\), the difference between characteristic functions of \(\int_Jf_td(\xi\theta_T^{-1})\) and \(\int_Jf_td(\eta \theta_T^{-1})\) converges uniformly to zero on every compact subset of \(\mathbb R\) as \(T\to \infty\). Therefore, there is hope that \(d({\mathcal L}(\xi\theta_T^{-1}| _J), {\mathcal L}(\eta\theta_T^{-1}| _J))\) can be shown to be small for large \(T\) if we choose for \(d\) a probability distance between distributions of point processes which metrizes a topology that is equal to or not too much finer than the weak topology (i.e., the topology of convergence in distribution). It was shown by A. D. Barbour, L. Holst, and S. Janson [“Poisson approximation” (1992; Zbl 0746.60002)] that \(d\) should be \(d_2\)-distance which can be defined by the formula \(d_2(P, Q) := \sup_{f\in F_2}\left| \int f\, dP - \int f\, dQ\right| \), where \(P\) and \(Q\) are probability measures on \(\mathcal M\), \(\mathcal M\) is the space of point measures on a compact set \(\mathcal X\), \(F_2\) is a certain class of functions determined on \(\mathcal M\).
In the present article under the above mentioned conditions \(1)-3)\), explicit upper bounds are given for the \(d_2\)-distance between the corresponding point process distributions. A number of related results, and applications to kernel density estimation and long range dependence testing are also presented.

Reviewer: Viktor Oganyan (Erevan)

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
62E20	Asymptotic distribution theory in statistics
62G07	Density estimation

Keywords:

Poisson process approximation; Stein’s method; density estimation; total variation distance

Citations:

Zbl 0285.60039; Zbl 0801.60027; Zbl 0609.60059; Zbl 0746.60002

Software:

spatial

References:

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