Properties of random stochastic matrices in the detection problem for a small signal-to-noise ratio

The article considers the problem of detecting a useful harmonic signal in noise for a small signal-to-noise ratio. A new detection mechanism based on the construction of a stochastic matrix based on a single signal window is proposed. The properties of the considered limit matrices and their limit vectors are investigated and established, for which the corresponding lemmas and theorems are formulated and proved. Formulas are obtained that relate these vectors to the noise envelope with known parameters.