a positive-definite function

Positive-definiteness is the precise analytic condition Bochner's theorem isolates as the signature of a characteristic function. It is also the difference-type special case of the positive-definite kernels that appear all over probability and analysis (covariance functions, reproducing kernels). Understanding it makes the cf method feel inevitable rather than magical.

A complex-valued function phi on R^d is positive-definite if for every n, every choice of points t_1, ..., t_n in R^d, and every complex c_1, ..., c_n, the sum sum_{j,k=1}^n c_j conj(c_k) phi(t_j - t_k) is real and nonnegative. Three immediate consequences follow purely from the definition: phi(0) >= 0 (take n=1); |phi(t)| <= phi(0) for all t; and phi(-t) = conj(phi(t)), so phi is Hermitian. Note the function depends only on the DIFFERENCE t_j - t_k, which is what distinguishes a positive-definite function from a general positive-definite kernel K(s,t); a stationary covariance K(s,t) = phi(s-t) is exactly a positive-definite function up to scaling.

Positive-definite functions matter because they bridge three worlds: Fourier transforms of positive measures (Bochner), characteristic functions of probability laws (when phi(0)=1 and phi is continuous), and covariance structures of stationary processes. A practical caveat: products of positive-definite functions are positive-definite (cf of independent sums), and pointwise limits of positive-definite functions stay positive-definite, but pointwise limits need NOT be continuous at 0 — and that loss of continuity is exactly the failure mode where a sequence of laws loses mass to infinity rather than converging.