Characteristic functions and convolution

For a real random variable X, define its characteristic function at a real frequency t as the expectation of exp(itX). The real part averages cosines and the imaginary part averages sines. Absolute integrability is automatic because the complex exponential has magnitude one. Thus the transform is defined at every real t, equals one at zero, and has absolute value at most one. This differs from a moment-generating function, which may diverge at every nonzero real argument.

Dominated convergence proves continuity: when the frequency changes by a small amount, the integrand converges pointwise and remains uniformly bounded. In fact the transform is uniformly continuous. Translation and scaling are especially transparent. Adding a constant rotates the transform by a deterministic phase, while multiplying X by a constant rescales the frequency. These rules concern the law of a variable and do not require a probability density.

If X and Y are independent, exp(itX) and exp(itY) are independent bounded variables. Their product is exp(it(X+Y)), so its expectation factors. At the measure level, the law of the sum is the convolution of the two laws: integrate the indicator that x plus y belongs to the desired event against the product measure. The transform changes this two-variable integration into an ordinary product of functions.

Independence is the hypothesis that licenses the product rule. Knowing the two marginal laws alone does not determine the sum when dependence is unknown. For instance, two symmetric signs can be equal or opposite; their individual transforms agree in both constructions, but their sums have different laws. If the second absolute moment exists, differentiating under the expectation gives the first two derivatives at zero and recovers the mean and second moment. Without the relevant integrability, formal differentiation is unjustified.

A probability law is uniquely determined by its characteristic function. Lévy’s continuity theorem makes this uniqueness useful for limits: if characteristic functions converge pointwise to a function continuous at zero, that limiting function is itself a characteristic function and the associated laws converge weakly. If the candidate is already known to be the transform of a probability law, continuity at zero is automatic. Pointwise convergence without that condition can describe probability mass escaping to infinity.

For a Bernoulli variable with success probability p, the transform is one minus p plus p exp(it). The transform of n independent copies is its nth power; expanding the polynomial recovers the binomial coefficients and probabilities. This is a useful two-way check: convolution predicts a sum law, while polynomial multiplication checks its normalization and coefficients. It also prepares the central limit theorem, where centering and scaling force the transform to be inspected near frequency zero.