Abstract:
We study the exact asymptotic of probabilities of high extrema of the Gaussian chaos processes, i.e. homogeneous function of a positive order from a Gaussian stationary vector process with dependent and non-identically distributed components. We assume that covariance matrix of the Gaussian stationary vector process satisfies Pickands' type condition in the neighbourhood of zero. In the first part of the thesis we consider the case of the product of two Gaussian stationary processes. In the second part the case of a quadratic form from a Gaussian stationary vector process was considered. It is assumed that the maximum eigenvalue of the quadratic form has multiplicity 1. In the third part of the thesis we consider the general case which generalizes the special cases considered above. We assume that the homogeneous function is a twice continuously differentiable in the neighbourhood of the set of maximum points of this function on the unit sphere. It turns out that the exact asymptotic of probability under investigation determined by the value of this maximum and by the structure of the set of maximum points of the homogeneous function on the unit sphere.
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