Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / M / M4 / a

The moment-generating function is $M_X(t)=\mathbb{E}[e^{tX}]$ for real $t$ where this expectation is finite. Usually one requires finiteness on a neighbourhood of zero when using it to characterize a distribution. The characteristic function isIt always exists because the integrand has absolute value one. In both definitions the expectation is taken with respect to the distribution of $X$.

A characteristic function is useful for proving convergence in distribution, including for laws without a finite moment-generating function: the continuity theorem says that pointwise convergence of characteristic functions to a characteristic function continuous at zero implies convergence in distribution to its law. For example, the Cauchy distribution has a characteristic function although its moment-generating function is not finite on any neighbourhood of zero.

A moment-generating function is especially useful for exponential tail bounds: for $t>0$ where it is finite, Markov's inequality givesOptimizing over $t$ yields a Chernoff bound. The positive real exponential in this argument permits the inequality; a unit-modulus complex exponential does not give the corresponding tail bound.