Characteristic Functions and the Berry-Esseen Theorem
This chapter discusses the role of the characteristic function in describing probability distributions. Theorems allowing the underlying probability function to be reconstructed from the characteristic function are presented. Results are also derived outlining the sense in which inversion of an approximate characteristic function leads to an approximate density or distribution function. These results are applied to derive Berry-Esseen theorems quantifying the error incurred in such an approximation. Finally, the relation between the characteristic function and moments and cumulants are investigated.
KeywordsCharacteristic Function Cumulative Distribution Function Moment Generate Function Saddlepoint Approximation Cumulant Generate Function
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