a probability measure μ. Chasing through the isomorphisms then gives On the other hand, given a probability measure μ on , the function is a normalized continuous positive-definite function. Continuity of f follows from the dominated convergence theorem. For positive-definiteness, take a nondegenerate representation of . This extends uniquely to a representation of its multiplier algebra and therefore a strongly continuous unitary representation Ug. As above we have f given by some vector state on Ug therefore positive-definite. The two constructions are mutual inverses. Special cases Bochner's theorem in the special case of the discrete group Z is often referred to as Herglotz's theorem (see Herglotz representation theorem) and says that a function f on Z with f(0) = 1 is positive-definite if and only if there exists a probability measure μ on the circle T such that Similarly, a continuous function f on R with f(0) = 1 is positive-definite if and only if there exists a probability measure μ on R such that Applications In statistics, Bochner's theorem can be used to describe the serial correlation of certain type of time series. A sequence of random variables of mean 0 is a (wide-sense) stationary time series if the covariance only depends on n − m. The function is called the autocovariance function of the time series. By the mean zero assumption, where ⟨⋅, ⋅⟩ denotes the inner product on the Hilbert space of random variables with finite second moments. It is then immediate that g is a positive-definite function on the integers . By Bochner's theorem, there exists a unique positive measure μ on [0, 1] such that This measure μ is called the spectral measure of the time series. It yields information about the "seasonal trends" of the series. For example, let z be an m-th root of unity (with the current identification, this is 1/m ∈ [0, 1]) and f be a random variable of mean 0 and variance 1. Consider the time series . The autocovariance function is Evidently, the corresponding spectral measure is the Dirac point mass centered at z. This is related to the fact that the time series repeats itself every m periods. When g has sufficiently fast decay, the measure μ is absolutely continuous with respect to the Lebesgue measure, and its Radon–Nikodym derivative f is called the spectral density of the time series. When g lies in , f is