On the Theory of Multidimensional Stationary Random Processes (1941): Translation and Commentary

Bulat Gafarov and Madina Karamysheva

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Abstract: This edition provides an English translation and commentary on V. N. Zasukhin's 1941 note "On the Theory of Multidimensional Stationary Random Processes," originally published in Doklady Akademii Nauk SSSR, 33(7-8), 435-437. The note states a multivariate extension of Wold's decomposition: an orthogonal decomposition into singular and regular components, a one-sided moving-average representation of the regular component in terms of orthonormal innovation sequences, and a spectral criterion for regularity of full rank. The latter combines absolute continuity of the spectral functions with positivity and logarithmic integrability of the determinant of the spectral density matrix. Whittle (1953) explicitly credited Zasukhin with the multivariate generalization. The original results are presented without detailed proofs. A separate editorial note discusses their historical reception and explains the terminology. A retypeset Russian transcription is included for comparison.

Date: 2026-11
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