This book is intended to give an account of the theory of infi­ nitely divisible statistical experiments which started from LeCam, 1974. It includes a presentation of LeCam's basic results as well as new developments in the field. The book consists of four chapters written by different authors. Chapters I, III and IV have been prepared in Bayreuth with the support of the Deutsche Forschungsgemeinschaft (DFG); Chapter II is part of its author's Habilitationsschrift, 1982 (Dortmund). For the reader's convenience, the chapters have been unified in presentation, without neglecting differences in the individual styles of writing. The authors are grateful to Dr. C. Becker for carefully reviewing the manuscript. Furthermore, acknowledgements are gratefully extended to the DFG for partly subsidizing Dr. Becker and the second author by a grant. Some special words of thanks are due to Mrs. Witzigmann, who typed the final manuscript and its predecessors with patience and skill. Universitat Bayreuth und A. Janssen Universitat Dortmund, H. Milbrodt Dezember 1984 H. Strasser CONTENTS Preface Introduction L~its of Triangular Arrays of 14 I. EXEeriments (H. Milbrodt and H. Strasser) 1. Basic Concepts 14 19 2. Gaussian Exper~ents 3. Introduction to Poisson Experiments 25 4. Convergence of Poisson Experiments 32 5. Convergence of Triangular Arrays 38 6. Identification of Limit Experiments 47 The Levy-Khintchine Formula for Infinitely 55 II.

I. Limits of Triangular Arrays of Experiments.- 1. Basic Concepts.- 2. Gaussian Experiments.- 3. Introduction to Poisson Experiments.- 4. Convergence of Poisson Experiments.- 5. Convergence of Triangular Arrays.- 6. Identification of Limit Experiments.- II. The Lévy-Khintchine Formula for Infinitely Divisible Experiments.- 7. Preliminaries.- 8. Infinitely Divisible Probability Measures.- 9. The Lévy-Khintchine Formula for Standard Measures.- 10. The Lévy-Khintchine Formula for Arbitrary Regular Infinitely Divisible Statistical Experiments.- III. Representation of Poisson Experiments.- 11. Generalized Poisson Processes.- 12. Standard Poisson Experiments.- IV. Statistical Experiments with Independent Increments.- 13. Preliminaries.- 14. Experiments with Independent Increments.- 15. Existence and Construction of Experiments with Independent Increments.- 16. Infinitely Divisible Experiments with Independent Increments.- 17. Weak Convergence of Triangular Arrays to Experiments with Independent Increments.- 18. The Likelihood Process.- 19. Application to Densities with Jumps.- List of Symbols.- Author Index.