Characteristic Functions And Their Basic Properties

Why Use a Characteristic Function?

A moment generating function may fail to exist away from zero: the expectation of an exponential can be infinite. A characteristic function avoids that obstacle by using a complex exponential whose modulus is always one. It is therefore defined at every real argument for every real-valued random variable. Its elementary properties—boundedness, continuity, its value at zero, and conjugate symmetry—provide basic checks for calculations and a foundation for later uses of transforms.

Let \(X:\Omega\to\mathbb{R}\) be a real-valued random variable on a probability space \((\Omega,\mathcal{F},\mathbb{P})\). Here \(i\) denotes the imaginary unit, with \(i^2=-1\), and \(t\) is a real number. For a complex-valued random variable with integrable real and imaginary parts, expectation is defined by taking the expectations of those parts separately.

The defining expression is an average of points on the unit circle in the complex plane. The average need not itself have modulus one, but it cannot have modulus greater than one. The next result collects the basic properties and proves each directly from the definition.

Proof. First, for any integrable complex-valued random variable \(Z\), \(|\mathbb{E}Z|\le\mathbb{E}|Z|\). To verify this inequality, put \(w=\mathbb{E}Z\). If \(w=0\), the inequality follows from \(\mathbb{E}|Z|\ge0\). If \(w\ne0\), let \(u=\overline{w}/|w|\), so \(|u|=1\) and \(uw=|w|\). Consequently, \(|w|=\operatorname{Re}(\mathbb{E}[uZ])=\mathbb{E}[\operatorname{Re}(uZ)]\le\mathbb{E}|uZ|=\mathbb{E}|Z|\). Apply this with \(Z=e^{itX}\). Its modulus is one, so \(|\varphi_X(t)|\le\mathbb{E}1=1\).

For continuity, fix \(t\in\mathbb{R}\) and let \(s\) approach \(t\). For real \(a,b\), the inequality \(|e^{ia}-e^{ib}|\le|a-b|\) follows, for example, by integrating the derivative \(ie^{ix}\), whose modulus is one, between \(a\) and \(b\). Also, the difference between two points of modulus one has modulus at most two. Given \(\varepsilon>0\), choose \(K\ge1\) so that \(\mathbb{P}(|X|>K)<\varepsilon/4\). Such a \(K\) exists because the events \(\{|X|>m\}\), for positive integers \(m\), decrease to a probability-zero event when \(X\) is real-valued, and probability is continuous for decreasing events. Splitting according to \(|X|\le K\) and \(|X|>K\) gives

The second term is less than \(\varepsilon/2\). If \(|t-s|<\varepsilon/(2K)\), the first term is less than \(\varepsilon/2\), so \(|\varphi_X(t)-\varphi_X(s)|<\varepsilon\). This proves continuity at \(t\); since \(t\) was arbitrary, it proves continuity everywhere.

Finally, \(e^{i0X}=1\), so \(\varphi_X(0)=\mathbb{E}1=1\). For real \(tX\), \(e^{-itX}=\overline{e^{itX}}\). Expectation commutes with complex conjugation, as follows by writing a complex random variable as its real part plus \(i\) times its imaginary part. Therefore \(\varphi_X(-t)=\mathbb{E}[\overline{e^{itX}}]=\overline{\mathbb{E}[e^{itX}]}=\overline{\varphi_X(t)}\). This proves all four claims. \(\square\)

The continuity proof is useful beyond this particular transform: it does not assume that \(X\) has a density, finite moments, or bounded support. The tail probability controls the part where \(X\) may be large, while on a bounded range of \(X\), changing \(t\) slightly changes the exponential by a uniformly small amount.

Worked Calculations

Interpreting the Basic Properties

The bound \(|\varphi_X(t)|\le1\) is a general constraint, not a claim that the function is real or nonnegative. The two-point example has a complex-valued expression for general \(t\), and the density example has a nonzero imaginary part at \(t=\pi\). Conjugate symmetry describes how the values at opposite arguments are related: knowing \(\varphi_X(t)\) determines \(\varphi_X(-t)\), but it does not generally make either value real.

The value at zero is a useful normalization check. Any proposed characteristic function of a probability distribution must equal one there. Continuity is another necessary check, including at zero; an algebraic expression obtained for \(t\ne0\) may have a removable singularity at zero, so its value there should be evaluated from the defining expectation or established by a limit. The density example illustrates why the definition remains the reliable starting point at a special argument.

Common mistake: moving expectation into an exponential. In general, \(\mathbb{E}[e^{itX}]\) is not \(e^{it\mathbb{E}X}\). For the two-point variable, \(\mathbb{E}X=(-1)(\tfrac13)+2(\tfrac23)=-\tfrac13+\tfrac43=1\). At \(t=\pi\), the incorrect expression would give \(e^{i\pi\mathbb{E}X}=e^{i\pi}=-1\). The correct computation is \(\varphi_X(\pi)=\tfrac13(-1)+\tfrac23(1)=\tfrac13\). The mistake replaces the average of the exponentials by the exponential of the average, which is not a valid rule for expectation.

These basic properties are practical diagnostics. When calculating a transform, check that its value at zero is one, its modulus never exceeds one, and its values at \(t\) and \(-t\) are conjugates. Such checks can expose sign errors or missing probability weights. They are necessary properties; by themselves, however, they are not a proof that an arbitrary function is the characteristic function of a probability distribution.