Central limit theorems for empirical product densities of stationary point processes [Elektronische Ressource] / von Stella Veronica David

Central limit theorems
for empirical product densities
of stationary point processes
Dissertation
zur Erlangung des Doktorgrades
der Mathematisch-Naturwissenschaftlichen Fakulta¨t
der Universita¨t Augsburg
von
Stella Veronica David
Juli 2008Erstgutachter: Professor Dr. Lothar Heinrich, Universita¨t Augsburg
Zweitgutachter: Professor Dr. Volker Schmidt, Universita¨t Ulm
Tag der mu¨ndlichen Pru¨fung: 22. Dezember 2008Acknowledgements
Writing this dissertation would not have been possible without the help of many
peopletowhomIwouldlike toexpressmygratitude. FirstandforemostIwouldlike
to thank my supervisor Professor Dr. Lothar Heinrich for suggesting this interesting
topic, his guidance, and the hours we spent discussing my questions. I am deeply
indebted to him for his constant support and encouragement. I am very grateful to
Professor Dr. Volker Schmidt for fruitful discussions and for being a referee for this
thesis. I am also much obliged to Professor Dr. Friedrich Pukelsheim for his interest
in my work and for encouraging me through the time of my studies. I would like to
thank my dear colleagues for an inspiring and friendly atmosphere. In particular I
wish to thank Dr. habil. Ute Hahn for arousing my interest in point processes and
for lively discussions. Thanks also to the Deutsche Forschungsgemeinschaft for the
financialsupportIreceivedthroughtheprojectAsymptotik von Diskrepanzmaßen fu¨r
Charakteristiken zweiter Ordnung von ra¨umlichen Punktprozessen mit Anwendungen
zur Modellidentifikation for the last two years. The warmest thank you goes to my
family and friends for their supportand love, and to Thomas for his encouragement,
for his understanding, and for always being there for me.Abstract
In the present work we investigate kernel-type estimators for product densities and
for the pair correlation function for stationary spatial point processes. In the set-
ting of Brillinger-mixing point processes we present central limit theorems for these
estimators and for the integrated squared error of the estimators for the second-
order productdensity and thepair correlation function. Based on these central limit
theorems we can construct asymptotic goodness-of-fit tests for the distribution of a
stationary point process.
Zusammenfassung
In dieser Arbeit untersuchen wir Kernschatzer fur Produktdichten und fur die Paar-¨ ¨ ¨
korrelationsfunktion fu¨r station¨are ra¨umliche Punktprozesse. Im Fall von Brillinger-
mischenden Punktprozessen leiten wir fur diese Schatzer und fur den integrierten¨ ¨ ¨
quadratischen Fehler der empirischen Produktdichtezweiter Ordnungundder empi-
rischen Paarkorrelationsfunktion Zentrale Grenzwerts¨atze her. Aus diesen Zentralen
Grenzwertsatzen lassen sich Anpassungstests zur Prufung auf die Verteilung eines¨ ¨
stationa¨ren Punktprozesses konstruieren.
AMS Mathematics Subject Classification: primary:60G55,62M30; secondary:62G20Contents
Notation v
1 Introduction 1
2 Point processes 5
2.1 Definition of point processes, stationarity, and isotropy . . . . . . . . . . . . . . . 5
2.2 Moment measures and kth-order stationarity . . . . . . . . . . . . . . . . . . . . 6
2.3 Point process characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Examples of point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Mixing properties 19
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Implications of mixing properties on second-order characteristics . . . . . . . . . 21
3.3 Relations between mixing properties . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Examples for Brillinger-mixing point processes . . . . . . . . . . . . . . . . . . . 23
4 Cumulants 31
4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5 Estimators for product densities and the pair correlation function 41
5.1 Kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Estimators for product densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
i5.3 Estimators for the pair correlation function . . . . . . . . . . . . . . . . . . . . . 47
5.4 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 CLTs for empirical product densities and the empirical pair correlation func-
tion 51
6.1 CLTs for the empirical second-order product density and the empirical pair cor-
relation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.1.1 Asymptotic representation of the mean and the variance of the estimators 52
6.1.2 Central limit theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2 CLTs for empirical product densities of higher order . . . . . . . . . . . . . . . . 69
6.2.1 Asymptotic representation of the mean and the variance of the estimators 69
6.2.2 Central limit theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7 CLTs for the integrated squared error of the empirical second-order product
density and the empirical pair correlation function 77
7.1 CLTsfortheintegrated squarederrorofthesecond-orderproductdensityestimator 77
7.1.1 The integrated squared error and some preliminary considerations . . . . 78
7.1.2 Asymptotic representation of the mean and the variance . . . . . . . . . . 86
7.1.3 Central limit theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.2 CLTs for the integrated squared error of the pair correlation function estimator . 104
7.2.1 Asymptotic representation of the mean and the variance . . . . . . . . . . 105
7.2.2 Central limit theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8 Asymptotic goodness-of-fit tests 113
9 Summary and outlook 115
A Properties of the sequence of observation windows 117
Bibliography 121
iiList of Figures
2.1 Interpretation of theK-function, the pair correlation function, and product den-
2sities for point processes inR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
22.2 Simulated realization of a stationary Poisson process inR . . . . . . . . . . . . . 12
22.3 Simulated realization of a stationary Poisson cluster process inR . . . . . . . . 14
5.1 Interpretation of estimated pair correlation function and second-order product
density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.1 Example to Lemma A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
iiiiv