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Abstract In this work, an explicit expression is obtained for the distribution function of a centered and normalized data vector containing a deterministic signal (or without a signal) and Gaussian white noise with constant variance. The obtained representation is exact and does not rely on any asymptotics for a large number of observations in particular; it is ultimately expressed through an error function and elementary functions and requires iterative calculations based on a recurrence formula.
Baluev et al. (Wed,) studied this question.