FRACTIONAL POISSON PROCESS IN TERMS OF ALPHA-STABLE DENSITIES by DEXTER ODCHIGUE CAHOY Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Dissertation Advisor: Dr. Wojbor A. Woyczynski Department of Statistics CASE WESTERN RESERVE UNIVERSITY August 2007 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of DEXTER ODCHIGUE CAHOY candidate for the Doctor of Philosophy degree * Committee Chair: Dr. Wojbor A. Woyczynski Dissertation Advisor Professor Department of Statistics Committee: Dr. Joe M. Sedransk Professor Department of Statistics Committee: Dr. David Gurarie Professor Department of Mathematics Committee: Dr. Matthew J. Sobel Professor Department of Operations August 2007 *We also certify that written approval has been obtained for any proprietary material contained therein. Table of Contents Table of Contents . iii List of Tables . v List of Figures . vi Acknowledgment . viii Abstract . ix 1 Motivation and Introduction 1 1.1 Motivation . 1 1.2 Poisson Distribution . 2 1.3 Poisson Process . 4 1.4 α-stable Distribution . 8 1.4.1 Parameter Estimation . 10 1.5 Outline of The Remaining Chapters . 14 2 Generalizations of the Standard Poisson Process 16 2.1 Standard Poisson Process . 16 2.2 Standard Fractional Generalization I . 19 2.3 Standard Fractional Generalization II . 26 2.4 Non-Standard Fractional Generalization . 27 2.5 Fractional Compound Poisson Process . 28 2.6 Alternative Fractional Generalization . 29 3 Fractional Poisson Process 32 3.1 Some Known Properties of fPp . 32 3.2 Asymptotic Behavior of the Waiting Time Density . 35 3.3 Simulation of Waiting Time . 38 3.4 The Limiting Scaled nth Arrival Time Distribution . 41 3.5 Intermittency . 46 3.6 Stationarity and Dependence of Increments . 49 3.7 Covariance Structure and Self-Similarity . 54 3.8 Limiting Scaled Fractional Poisson Distribution . 58 3.9 Alternative fPp . 63 iii 4 Estimation 67 4.1 Method of Moments . 67 4.2 Asymptotic Normality of Estimators . 71 4.3 Numerical Experiment . 76 4.3.1 Simulated fPp Data . 77 5 Summary, Conclusions, and Future Research Directions 80 5.1 Summary . 80 5.2 Conclusions . 81 5.3 Future Research Directions . 81 Appendix 83 Appendix A. Some Properties of α+−Stable Densities . 83 Appendix B. Scaled Fractional Poisson Quantiles (3.12)....... 86 Bibliography 90 iv List of Tables 3.1 Properties of fPp compared with those of the ordinary Poisson process. 33 3.2 χ2 Goodness-of-fit Test Statistics with µ = 1. .............. 41 3.3 Parameter estimates of the fitted model atb, µ = 1. ........... 54 4.1 Test statistics for comparing parameter (ν, µ) = (0.9, 10) estimators using a simulated fPp data. ........................ 78 4.2 Test statistics for comparing parameter (ν, µ) = (0.3, 1) estimators using a simulated fPp data. ........................ 78 4.3 Test statistics for comparing parameter (ν, µ) = (0.2, 100) estimators using a simulated fPp data. ........................ 78 4.4 Test statistics for comparing parameter (ν, µ) = (0.6, 1000) estimators using a simulated fPp data. ........................ 79 5.1 Probability density (3.12) values for ν = 0.05(0.05)0.50 and z = 0.0(0.1)3.0. 86 5.2 Probability density (3.12) values for ν = 0.05(0.05)0.50 and z = 3.1(0.1)5.0. 87 5.3 Probability density (3.12) values for ν = 0.55(0.05)0.95 and z = 0.0(0.1)3.0. 88 5.4 Probability density (3.12) values for ν = 0.55(0.05)0.95 and z = 3.1(0.1)5.0. 89 v List of Figures 3.1 The mean of fPp as a function of time t and fractional order ν.... 34 3.2 The variance of fPp as a function of time t and fractional order ν... 35 3.3 Waiting time densities of fPp (3.1) using µ = 1, and ν = 0.1(0.1)1 (log-log scale). ............................... 37 3.4 Scaled nth arrival time distributions for standard Poisson process (3.8) with n = 1, 2, 3, 5, 10, 30, and µ = 1. ................... 42 3.5 Scaled nth fPp arrival time distributions (3.11) corresponding to ν = 0.5, n = 1, 3, 10, 30, and µ = 1 (log-log scale). .............. 46 3.6 Histograms for standard Poisson process (leftmost panel) and fractional Poisson processes of orders ν = 0.9 & 0.5 (middle and rightmost panels). 48 3.7 The limit proportion of empty bins R(ν) using a total of B=50 bins. 49 3.8 Dependence structure of the fPp increments for fractional orders ν = 0.4, 0.6, 0.7, 0.8, 0.95, and 0.99999, with µ = 1. ............. 50 3.9 Distribution of the fPp increments on the sampling intervals a)[0, 600], b)(600, 1200], c) (1200, 1800], and (1800, 2400] corresponding to frac- tional order ν = 0.99999, and µ = 1. .................. 51 3.10 Distribution of the fPp increments on the sampling intervals a)[0, 600], b)(600, 1200], c) (1200, 1800], and (1800, 2400] corresponding to frac- tional order ν = 0.8, and µ = 1. ..................... 52 3.11 Distribution of the fPp increments on the sampling intervals a)[0, 600], b)(600, 1200], c) (1200, 1800], and (1800, 2400] corresponding to frac- tional order ν = 0.6, and µ = 1. ..................... 53 3.12 The parameter estimate b as a function of ν, with µ = 1 . 55 3.13 The function atb fitted to the simulated covariance of fPp for different fractional order ν, with µ =1. ..................... 56 3.14 Two-dimensional covariance structure of fPp for fractional orders a) ν = 0.25, b) ν = 0.5, c) ν = 0.75, and d) ν = 1, with µ = 1. ..... 57 3.15 Limiting distribution (3.12) for ν= 0.1(0.1)0.9 and 0.95, with µ =1. 63 3.16 Sample trajectories of (a) standard Poisson process, (b) fPp, and (c) the alternative fPp generated by stochastic fractional differential equa- tion (3.17), with ν = 0.5. ......................... 66 vi A α+-Stable Densities . 85 vii ACKNOWLEDGMENTS With special thanks to the following individuals who have helped me in the de- velopment of my paper and who have supported me throughout my graduate studies: Dr. Wojbor A. Woyczynski, for your guidance and support as my graduate ad- visor, and for your confidence in my capabilities that has inspired me to become a better individual and statistician; Dr. Vladimir V. Uchaikin, for the opportunity to work with you, and Dr. Enrico Scalas, for your help in clarifying some details in renewal theory; My panelists and professors in the department - Dr. Joe M. Sedransk, Dr. David Gurarie and Dr. Matthew J. Sobel. I am grateful for having you in my committee and for helping me improve my thesis with your superb suggestions and advice, Dr. Jiayang Sun for your encouragement in developing my leadership and research skills, and Dr. Joe M. Sedransk, the training I received in your class has significantly enhanced my research experience, and Dr. James C. Alexander and Dr. Jill E. Korbin for the graduate assistantship grant; Ms. Sharon Dingess, for your untiring assistance with my office needs throughout my tenure as a student in the department, and to my friends and classmates in the Department of Statistics; My parents, brother and sisters for your support of my goals and aspirations; The First Assembly of God Church, particularly the Multimedia Ministry, for providing spiritual shelter and moral back-up; Edwin and Ferly Santos, Malou Dham, Lowell Lorenzo, Pastor Clint and Sally Bryan, and Marcelo and Lynn Gonzalez for your friendship; My wife, Armi and my daughter, Ysa, for your love and prayers and the inspiration you have given me to be a better husband and father; Most of all, to my Heavenly Father who is the source of all good and perfect gift, and to my Lord and Savior Jesus Christ from whom all blessings flow, my love and praises. viii Fractional Poisson Process in Terms of Alpha-Stable Densities Abstract by Dexter Odchigue Cahoy The link between fractional Poisson process (fPp) and α-stable density is established by solving an integral equation. The result is then used to study the properties of fPp such as asymptotical n-th arrival time, number of events distributions, covariance structure, stationarity and dependence of increments, self-similarity, and intermit- tency property. Asymptotically normal parameter estimators and their variants are derived; their properties are studied and compared using synthetic data. An alterna- tive fPp model is also proposed.Finally, the asymptotic distribution of a scaled fPp random variable is shown to be free of some parameters; formulae for integer-order, non-central moments are also derived. Keywords: fractional Poisson process, α-stable, intermittency, scaled fPp, self- similarity ix Chapter 1 Motivation and Introduction 1.1 Motivation For almost two centuries, Poisson process served as the simplest, and yet one of the most important stochastic models. Its main properties, namely, absence of memory and jump-shaped increments model a large number of processes in several scientific fields such as epidemiology, industry, biology, queueing theory, traffic flow, and commerce (see Haight (1967, chap. 7)). On the other hand, there are many processes that exhibit long memory (e.g., network traffic and other complex systems) as well. It would be useful if one could generalize the standard Poisson process to include systems or processes that don’t have rapid memory loss in the long run. It is largely this appealing feature that drives this thesis to investigate further the sta- tistical properties of a particular generalization of a Poisson process called fractional Poisson process (fPp). Moreover, the generalization has some parameters that need to be estimated in order for the model to be applicable to a wide variety of interesting counting phe- nomena.