INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI fihns the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter 6ce, while others may be from any type of computer printer. The quality of this reproduction Is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedtfarough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not said UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, b^inning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Infiumadon Company 300 North Zeeb Road, Ann Aibor MI 48106-1346 USA 313/761-4700 800/521-0600 A CENTRAL LIMIT THEOREM FOR COMPLEX-VALUED PROBABILITIES DISSERTATION Presented in Partial FulfiUment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Natalia A. Humphreys, M.S. ***** The Ohio State University 1999 Dissertation Commitee: Approved by Professor Bogdan M. Baishanski, Advisor Professor Gerald A. Edgar Advisor Professor Paul G. Nevai Department Of Mathematics ÜMX Number: 9919871 Copyright 1999 by Humphreys, Natalia Alexandra All rights reserved. UMI Microform 9919871 Copyright 1999, by UMI Company. All rights reserved. This microform edition Is protected against unauthorized copying under Title 17, United States Code. m a 300 North Zeeb Road Ann Arbor, MI 48103 © Copyright by Natalia A. Humphreys 1999 ABSTRACT If Ç is a non-negative integrable function on R, satisfying some rather general con­ ditions. then the behavior of the n-fold convolutions ''----------V---------- ' n times n —>■ oc is described by the Central Limit Theorem. However, the problem of describ­ ing the behavior of rises in several contexts when <p is not a positive function. Central Limit Theorem was extended to such non-positive probabilities (signed or complex-valued) by a number of authors: A. Zhukov in numerical analysis, V. Krylov in partial differential equations, R. Hersh and K. Hochberg in distribution theory. In these extensions it was typically assumed that ip (the Fourier transform of p) satisfies the condition Ini^(t) = (A -l-o(l))i’. i —> 0, where A ^ 0, ç an integer > 2. and it Wtvs shown that converges in some weak sense to the inverse Fourier transform of exp(AC^)). Recently B. Baishanski has argued that in case Re A = 0, the scaling factor is not the "natural” one. Namely, it is known that in case of probability densities the scaling factor is essentially unique, so the “natural” scaling for in complex­ valued case should be the same as the essentially unique scaling of He has considered two examples and shown that under the natural scaling one obtains analogues of the Central Limit Theorem of a new kind when Re A = 0. We have obtained more general results of the same nature. Our main theorem is 11 Theorem . Suppose that ip is a complex-valued function such that p{u) e L^(R) n L"(R) for some s > 1 |<p(t)l < 1^(0) I = 1 for every t ^ 0 6 L \R ) ? [nip{t) = i ^2 4- G{t), t £ U — a neighborhood of zero , r= p where p, g G N, 2 < p < q, Op, Op+i , . , a,, /? are real, > 0, 3 > 0. G 6 C'^iU), G"{t) = 0 { t‘>-^), G{t) = t -)■ 0. Let the scaled n-fold convolutions be given by \/\u)x{c), where ipxic) = cr„>p^'‘^(cr„Ac), a„ = . A = Then \/Xib\ will exhibit one of two types of very regular divergence: I. If p is even, then V X ' i p x i c ) = F(c)ua(c) + ri(c), where F[c) is an integrable function given by F(c) = B\c\~'^ exp { -F lc l''} , (1) |ua(c)| = 1 and Q |rA(c)| < for |c| < In^ A, A > A , where (2) B, E, C, e,7 ,p and A are positive constants that depend on p and q. ui Moreover, the family (ua) satisfies: for every interval [a, 6] not containing zero and every arc I of the unit circle lim —m {c : c € [a. 6],ua(c) € /} =-^{ardength of I) A —t o o b — d 2 7 T 2. If p is odd. then \/Â|V'A(c)| = ^{c)tx{c) + rA(c). where [ 2F(c), i/o o , F{c) being defined by (1),^ < t\{c) < 1 andr\{c) satisfies (2). Moreover, the family (éa) satisfies: for every interval [a. 6j not containing zero and for every subinterval I of [0,1] lim —^ m { c : c G [a,6],ÉA(c) e 1} = - f , ^ — \ —*oo 0 — a JI \/\ — X As an application of our main Theorem we obtain a continuous analogue of Girard’s asymptotic formulae for norms of powers of absolutely convergent Fourier series. IV Dedicated to my family ACKNOWLEDGMENTS First and foremost, I would like to express my deep gratitude to my teacher. Pro­ fessor Bogdan M. Baishanski, for his helpful discussions and the suggestion of the topic for this dissertation. His intellectual support and warm encouragement were invaluable. His wisdom, patience, humor and kindness were much appreciated. I am especially grateful to him for his careful and detailed criticism of all my written work. I am thankful for his kind permission to use his recent work "Norms of Pow­ ers and a Central Limit Theorem for Complex-Valued Probabilities” as part of my Introduction. The works of D. M. Girard “A General Asymptotic Formula For Analytic Func­ tions” and of G. W. Hedstrom “Norms of Powers of Absolutely Convergent Fourier Series” were also of a great help in creating this dissertation. I would like to thank the members of my committee, Professors Gerald Edgar and Paul Nevai. for their time and patience in reading this work. Thanks to the OSU Mathematics Department faculty members who have taught me during the past years, as well as the department itself for academic and financial support. I am especially grateful to Professors Ranko Bojanic and Saleh Tanveer. VI I wish to express my deepest gratitude to my husband, John Humphreys, for his constant support, unbounded love and unwavering faith in me. I would like to thank my dear parents. Alexander and Vera Bondarev for their love, laughter and for giving me perspective throughout my time in graduate school. I am most grateful to all of my relatives and especially to my grandmother, Natalia Slepjan-Bondareva, for their love and for believing in me. I wish to thank my parents-in-law, Charles and Ann Humphreys, for their contin­ uing emotional support and kindness. Finally. I would like to express my deepest gratitude to my friends: Tona Dickerson and Thomas Stacklin for their vital encouragement. Without all of these people this project would not have been possible. vu VITA July 26, 1972 ........................................... Born - Leningrad (now St. Petersburg), USSR (now Russia). June. 1993 ................................................ M.S. Mathematics. Department of Mathematics and Mechanics. St. Petersburg State University, Russia. June. 1993 - .A.ugust. 1994 ....................... Graduate School Fellow. Department of Mathematics. The Ohio State University. September. 1994 - present ....................... Graduate Teaching .\ssistant. Department of Mathematics. The Ohio State Universitv. PUBLICATIONS Bondareva (Humphreys), N. A. and Bondarev, .A.. S.. (1998) Representations of the Functionals on the Regularly Normalized Operator Ideals, (Russian), Izvestiya GETU. Fundamental mathematical models and their application in electronics and automatics. 512: 13-24. Bondareva (Humphreys), N. A. and Bondarev, A. S., (1996) Lattice Nuclear and Lattice Integral Operators, (Russian), Izvestiya GETU. Collection of scientific works. Mathematics, 450: 10-18. vui Bondareva (Humphreys), N. A. and Bondarev, A. S., (1994) Absolutely Summing Operators, (Russian). Izvestiya GETU. Collection of scientific works. Mathematics. 472: 13-18. FIELDS OF STUDY Major field: Mathematics Specialization: Mathematical Analysis Studies in .-Vpplied Complex Variables Professor Saleh Tanveer .■Approximation Theory Professors Ranko Bojanic and Paul Nevai .A.symptotic Analysis Professor Saleh Tanveer Complex .Analysis Professor Francis Carroll Differential Equations Professor Saleh Tanveer Real Analysis Professors Gerald Edgar and Paul Nevai IX TABLE OF CONTENTS Abstract ii Dedication v Acknowledgments vi V ita viii CHAPTER PAGE 1 Introduction 1 1.1 The Classical Central Limit Theorem ...................................................... 1 1.2 Signed and complex-valued probabihties ................................................ 2 1.3 Our problem ................................................................................................... 3 1.4 Two cases ...................................................................................................... 4 1.5 Why the new scaling? ................................................................................ 8 1.6 Another kind of central limit theorems ................................................... 9 1.7 Norms of powers ......................................................................................... 19 1.8 Thesis structure ...........................................................................................