On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin

E. J. G. Pitman

doi:10.1017/S1446788700006121

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Let X be a real valued random variable with probability measure P and distribution function F. It will be convenient to take F as the intermediate distribution function defined by . In mathematical analysis it is a little more convenient to use this function rather than , which arise more naturally in probability theory. In all cases we shall consider With this definition, if the distribution function of X is F(x), then the distribution function of −X is 1−F(−x). The distribution of X is symmetrical about 0 if F(x) = 1 − F(−x).

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