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Khintchine: describing stationary fluctuations through a spectrum of frequencies

Contribution grade

Probability / stationary processes · Exact theorem and reusable mathematical method

Reference score300,000Provisional editorial estimate of this exact contribution; its grade is assessed separately.

What changed

A fluctuating signal may look irregular while its average and its pairwise correlation remain stable over time. Khintchine builds a theory for this second-order stationary setting. For a real normalized process with the required mean-square continuity, the correlation is a cosine average over a probability distribution of frequencies. He also constructs a process for any such correlation. The distribution can contain individual frequency atoms as well as continuous parts. Using this representation, he studies long-run averages and the separation of persistent oscillations from a continuous-spectrum remainder. Bochner's earlier Fourier representation is an explicit mathematical input.

Scientific scope

A fluctuating signal may look irregular while its average and its pairwise correlation remain stable over time. Khintchine builds a theory for this second-order stationary setting. For a real normalized process with the required mean-square continuity, the correlation is a cosine average over a probability distribution of frequencies. He also constructs a process for any such correlation. The distribution can contain individual frequency atoms as well as continuous parts. Using this representation, he studies long-run averages and the separation of persistent oscillations from a continuous-spectrum remainder. Bochner's earlier Fourier representation is an explicit mathematical input. Second-order stationarity is weaker than stationarity of the full joint distributions. Mean-square continuity does not mean that individual paths are continuous. A spectral measure need not have a density. The continuous-spectrum remainder has vanishing averaged squared correlation, not necessarily pointwise decay or mixing. Time-average convergence in mean square or probability does not by itself identify the ensemble mean or give an almost-sure ergodic theorem. Earlier Bochner, Koopman, von Neumann and Kolmogorov contributions are kept separate.

Evidence

The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Second-order stationarity is weaker than stationarity of the full joint distributions. Mean-square continuity does not mean that individual paths are continuous. A spectral measure need not have a density. The continuous-spectrum remainder has vanishing averaged squared correlation, not necessarily pointwise decay or mixing. Time-average convergence in mean square or probability does not by itself identify the ensemble mean or give an almost-sure ergodic theorem. Earlier Bochner, Koopman, von Neumann and Kolmogorov contributions are kept separate.

Contributors

Provisional WRS assessment, edition 2026-10-23.7. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.

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