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During last decades the study of probabilities of large extremes of Gaussian random processes has been intensively progressed. Several powerful methods were developed to get both asymptotic behaviors of the probabilities and uniform boundaries for them (for details and further references, see [1],[7],[2],[10],[3]). It turned out that some of them, in particular, the double-sum method could be also applied to Gaussian random fields, so that asymptotic methods for Gaussian random fields were developed in parallel with the methods for Gaussian processes (see [2],[10]). But, in spite of increasing interest to behaviors of large deviations of Gaussian vector processes (see [8],[2],[12]), one can see some backlog in development of corresponding asymptotic methods for the vector case. The present paper is in abreast of the works on generalizations of the double-sum method to Gaussian vector processes (see [9],[4],[10]). Along with [5], we consider the norm of a Gaussian vector cyclostationary processes with independent identically distributed components. We call this process a χ-process, which also is cyclostationary. A random process X (t), t∈ R, is called cyclostationary with period τ if, for every v∈ R, its mean EX (t) and covariance function rt (v)= r (t, t+ v) are periodic functions in t with period τ.

Results and references related to spectral properties of this class of random processes can be found in [13](see also [5]). A spectral representation and conditions for the harmonizability of a process are also provided there. The evolutionary spectral density can be estimated by means of the periodogram. Similar problems where considered in [11] for stationary χ …