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In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group.
For continuous time, the Wiener–Khinchin theorem says that if is a wide-sense-stationary random process whose autocorrelation function (sometimes called autocovariance) defined in terms of statistical expected value, () = [() ()] exists and is finite at every lag , then there exists a monotone function in the frequency domain < <, or equivalently a non negative Radon measure on the frequency ...
Bochner’s theorem. An arbitrary function φ : R n → C is the characteristic function of some random variable if and only if φ is positive definite , continuous at the origin, and if φ (0) = 1 .
Khinchin's theorem may refer to any of several different results by Aleksandr Khinchin: Wiener–Khinchin theorem; Khinchin's constant;
The Lévy–Khintchine representation for the symmetric Cauchy process is a triplet with zero drift and zero diffusion, giving a Lévy–Khintchine triplet of (,,), where () = / (). [8] The marginal characteristic function of the symmetric Cauchy process has the form: [1] [8]
The proof presented here was arranged by Czesław Ryll-Nardzewski [3] and is much simpler than Khinchin's original proof which did not use ergodic theory.. Since the first coefficient a 0 of the continued fraction of x plays no role in Khinchin's theorem and since the rational numbers have Lebesgue measure zero, we are reduced to the study of irrational numbers in the unit interval, i.e ...
Well, now that I've looked at Khintchine's original article, where he cites Bochner and relies on Bochner's theorem, I see that actually «independent» is really not the appropriate word. Khintchine proved the analogous result in the different context of stochastic processes, whereas Wiener did it for the sample functions.
Recall that it follows from Lusin's theorem that a Lebesgue-measurable function is approximately continuous almost everywhere (and conversely). [ 10 ] [ 11 ] The key theorem in constructing the Khinchin integral is this: a function f that is generalized absolutely continuous (or even of "generalized bounded variation", a weaker notion) has an ...