Bochner's theorem
/ BOKH-ner /
The characteristic function (cf) phi(t) = E[e^(i t X)] turns a probability law on the real line into a single bounded function of a real (or vector) frequency t. A natural question sits at the foundation of Fourier-analytic probability: which functions phi are characteristic functions of some probability measure? Bochner's theorem gives the exact answer, and it is the structural backbone behind Levy's continuity theorem and the whole cf method.
Bochner's theorem says that a continuous function phi on R^d with phi(0) = 1 is the characteristic function of some probability measure if and only if phi is positive-definite. Positive-definite means: for every finite set of frequencies t_1, ..., t_n and every complex numbers c_1, ..., c_n, the Hermitian sum sum_{j,k} c_j conj(c_k) phi(t_j - t_k) >= 0. Equivalently the matrix with entries phi(t_j - t_k) is positive-semidefinite. The 'only if' direction is a quick computation: plug phi(t) = E[e^(i t X)] into the sum and it becomes E[ |sum_j c_j e^(i t_j X)|^2 ] >= 0. The 'if' direction is the deep part: positive-definiteness plus continuity at 0 forces a (nonnegative, total-mass-1) measure to exist via Fourier inversion.
Bochner's theorem matters because it converts an analytic property (positive-definiteness, checkable on paper) into a probabilistic existence statement, and it is the abstract reason cf arguments work. It generalises far beyond R^d: on a locally compact abelian group the positive-definite functions are exactly the Fourier transforms of finite positive measures on the dual group, which underlies stationary-process spectral theory. One honest caveat: continuity at the origin is essential. A function can be positive-definite as a pointwise condition yet fail to be a cf if it is discontinuous at 0; continuity at 0 is what guarantees the candidate measure has total mass 1 rather than leaking mass to infinity.
The function phi(t) = e^(-|t|) is continuous, equals 1 at 0, and is positive-definite, so by Bochner it must be a cf — indeed it is the cf of the standard Cauchy distribution. By contrast phi(t) = cos(t^2) equals 1 at 0 and is continuous but is NOT positive-definite (one can find frequencies t_j making the Hermitian sum negative), so no random variable has it as a characteristic function.
Positive-definiteness, not mere boundedness or phi(0)=1, is what makes a function a characteristic function.
A common error is to think any continuous phi with phi(0)=1 and |phi|<=1 is a cf; you also need positive-definiteness, and dropping continuity at 0 lets total mass escape to infinity.