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Probability and Stochastic Processes by Ionut Florescu

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6.4 Characteristic Function

The characteristic function of a random variable is a powerful tool for analyzing the distribution of sums of independent random variables. In the same way in which the MGF was related to the Laplace transform the characteristic function is related to the Fourier transform of a function.

Since c06-math-0174 for every c06-math-0175 and c06-math-0176 it is clear that the expectation defined above exists and

equation

This is easy to prove using the Jensen inequality (Lemma 4.14) for the convex function c06-math-0178.

If X has a probability density then the characteristic ...

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