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Probability Characteristics of the Absolute Maximum of the Discontinuous Homogeneous Gaussian Random Field

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A new method is introduced in order to determine the limiting characteristics of the homogeneous Gaussian random field with the non-differentiable correlation function at zero. This method is based on the representation of a field in the small neighborhood of the maximum point by the superposition of the statistically independent locally Markov Gaussian random processes. Then, using the expressions for the distribution function of the Markov Gaussian process, it is possible to obtain the asymptotic approximations for both the probability of the threshold crossing by the absolute maximum of the homogeneous Gaussian random field and the distribution function of the absolute maximum of the field. It is shown that the proposed approximations are monotonic and valid in a much wider range of both threshold values and the area of the definitional domain of the random field, unlike the commonly known approximations. The applicability borders of the introduced theoretical formulas are established by means of the statistical computer simulation.
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Discontinuous Random Field; Distribution Function of the Absolute Maximum; Local Markov Approximation; Statistical Simulation

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