By Arnold Janssen
This booklet is meant to provide an account of the idea of infi nitely divisible statistical experiments which began from LeCam, 1974. It incorporates a presentation of LeCam's easy effects in addition to new advancements within the box. The e-book contains 4 chapters written through varied authors. Chapters I, III and IV were ready in Bayreuth with the help of the Deutsche Forschungsgemeinschaft (DFG); bankruptcy II is a part of its author's Habilitationsschrift, 1982 (Dortmund). For the reader's comfort, the chapters were unified in presentation, with out neglecting ameliorations within the person types of writing. The authors are thankful to Dr. C. Becker for conscientiously reviewing the manuscript. moreover, acknowledgements are gratefully prolonged to the DFG for in part subsidizing Dr. Becker and the second one writer via a furnish. a few distinct phrases of thank you are because of Mrs. Witzigmann, who typed the ultimate manuscript and its predecessors with persistence and ability. Universitat Bayreuth und A. Janssen Universitat Dortmund, H. Milbrodt Dezember 1984 H. Strasser CONTENTS Preface advent L its of Triangular Arrays of 14 I. EXEeriments (H. Milbrodt and H. Strasser) 1. easy recommendations 14 19 2. Gaussian Exper ents three. advent to Poisson Experiments 25 four. Convergence of Poisson Experiments 32 five. Convergence of Triangular Arrays 38 6. id of restrict Experiments forty seven The Levy-Khintchine formulation for Infinitely fifty five II.
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L dP nit dP niu dP . u 1 )2 TaT ( . dP niq dP nit en;-nit dP nit . <4<:} dP . r Hence, (PI) follows from the condition stated in the theorem. CI Different to the Gaussian case, it is not possible to detect a Poisson experiment from its binary subexperiments, in general. 5) cannot be expected to hold. This was one of the basic motivations for introducing the concept of experiments with independent increments (cf. the last contribution in this volume). Before determining those "mixed cases" in which only compound Poisson factors can occur, we give some examples.
A. a. } = 0 and f Wz dMa. < 00, z E Sa. This was shown in the preceding proof. The restriction M {e } = 0 is imposed for convenience only; 2 a. a. since sa. ' it is without any influence. 6) Definition. An experiment E for the parameter space T is a Poisson experiment if there are Levy measures M I B , a. E A(T) , a. a that zES a such a EA(T). This terminology will be justified in Part III. There it is shown that it coincides with the one introduced by LeCam, 1974: Every Poisson experiment is equivalent to an experiment consisting of distributions of generalized Poisson processes and vice versa.
Part II, Sec. 10. 47 6. e. a triangular array Eni (Qni' Ani' {P nit : tET}), 1 ~i~kn' experiments for some parameter space nElN, of T and the appertaining sequence (En) n E IN of product experiments. ) a. ) 1 5: . ,;; k E IN' Throughout this whole section n1. - 1. - n' n we shall assume that (E ni ) 1 ~ i ;;;; k , n E IN is infinitesimal and bounded. n We give conditions under which all weak accumulation points of (En) n E IN are either Gaussian or Poisson experiments. 1) Definition. ) 1 ;;;; .