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

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.

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### Software:

spatial### References:

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[2] | Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Univ. Press. · Zbl 0746.60002 |

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[5] | Daley, D. J. (1974). Various concepts of orderliness for point processes. In Stochastic Geometry (E. F. Harding and D. G. Kendall, eds.) 148–161. Wiley, New York. · Zbl 0285.60039 |

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[8] | Ellis, S. P. (1986). A limit theorem for spatial point processes. Adv. in Appl. Probab. 18 646–659. · Zbl 0609.60059 · doi:10.2307/1427181 |

[9] | Ellis, S. P. (1991). Density estimation for point processes. Stochastic Process. Appl. 39 345–358. · Zbl 0749.62023 · doi:10.1016/0304-4149(91)90087-S |

[10] | Kallenberg, O. (1986). Random Measures , 4th ed. Academic Press, New York. · Zbl 0345.60032 |

[11] | Kingman, J. F. C. (1993). Poisson Processes . Oxford Univ. Press. · Zbl 0771.60001 |

[12] | Rachev, S. T. (1984). The Monge–Kantorovich mass transference problem and its stochastic applications. Theory Probab. Appl. 29 647–676. · Zbl 0581.60010 · doi:10.1137/1129093 |

[13] | Ripley, B. D. (1981). Spatial Statistics . Wiley, New York. · Zbl 0583.62087 |

[14] | Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis . Chapman and Hall, London. · Zbl 0617.62042 |

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