A survey of the theory of characteristic functions

The paper gives a survey of the theory of univariate characteristic functions. These functions were originally introduced as tools in the study of limit theorems but it was later realized that they had an independent mathematical interest. Those parts of the theory which can be found in textbooks are treated only briefly; the main emphasis is placed on more recent developments and areas where active research is still in progress. UNIQUENESS, INVERSION, CONVOLUTION AND CONTINUITY THEOREMS; INFINITE CONVOLUTION; ANALYTIC AND BOUNDARY C. F.'S; DECOMPOSITION OF C.F.'S; INDECOMPOSABLE C.F.'S; INFINITELY DIVISIBLE C.F.'S; STABLE LAWS; THE CLASS Io; ALPHA DECOMPOSITIONS; CHARACTERIZATION PROBLEMS; STABILITY THEOREMS CONTENTS Page 1. DEFINITION AND FUNDAMENTAL PROPERTIES OF CHARACTERISTIC FUNCTIONS . . .. .. .. .. .. .. .. .. .. .. 2 2. BASIC THEOREMS FOR CHARACTERISTIC FUNCTIONS . . . 6 3. ANALYTIC CHARACTERISTIC FUNCTIONS AND BOUNDARY CHARACTERISTIC FUNCTIONS .. .. .. .. .. .. .. . .. 12 4. DECOMPOSITION OF CHARACTERISTIC FUNCTIONS .... .. 16 5. INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS .. .. 18 6. CLASSICAL THEOREMS FROM THE ARITHMETIC OF DISTRIBUTION FUNCTIONS ...... . 24 7. INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS WHICH HAVE ONLY INFINITELY DIVISIBLE FACTORS . . .. .. . .. 27 8. ALPHA-DECOMPOSITIONS .. 30 9. SOME REMARKS CONCERNING CHARACTERIZATION PROBLEMS .. 31 10. STABILITY THEOREMS ... ...... 32 11. REFERENCES ... 35 12. ADDENDUM ... . 37 Received 13 May 1971. The preparation of this paper was supported by the National Science Foundation, Grani NSF-9396.

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