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This work continues the research started by Rudzkis (Soviet Math Dokl, 45(1), 226–228, 1992), Rudzkis and Bakshaev (Lithuanian Mathematical Journal, 52(2), 196–213, 2012) and extends it to the case of random fields close to Gaussian ones. Let { ξ (t) , t ∈ ℝ m } be a differentiable (in the mean square sense) random field with E ξ (t) ≡ 0 , D ξ (t) ≡ 1 and continuous trajectories. The paper is devoted to the problem of large excursions of the random field ξ. Let T be an m-dimensional interval and u(t) be a continuously differentiable function. We investigate the asymptotic properties of the probability P = ℙ { ξ (t) < u (t) , t ∈ T } as inf t ∈ ℝ m u (t) → ∞ and the mixed cumulants of the random field ξ and its partial derivatives tend to zero, i.e. the scheme of series is considered. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then 1 − P 1 − G = 1 + o (1) , where G is a constructive functional depending on u, T and a matrix function R (t) = c o v (ξ ′ (t) , ξ ′ (t)) , ξ ′ (t) = ∂ ξ (t) ∂ t 1 ,... , ∂ ξ (t) ∂ t m . [ABSTRACT FROM AUTHOR]

In the paper, we present some general theorems on large deviations of random vectors with cumulants satisfying the generalized Statulevičius condition. The results obtained are applicable in derivation of limit theorems in the scheme of series, including the case where the dimension of the considered random vectors is growing indefinitely. [ABSTRACT FROM AUTHOR]

The paper is devoted to the supremum-type multivariate goodness-of-fit tests based on the empirical characteristic function. Particular attention is devoted to the composite hypothesis of normality and Gaussian distribution mixture model. An analytical way to approximate the null asymptotic distributions of the considered test statistics is discussed applying the theory of large excursions of differentiable Gaussian random fields. The produced comparative Monte Carlo power study shows that the considered tests are powerful competitors to the existing classical criteria, clearly dominating in verification of the goodness-of-fit hypotheses against the specific types of alternatives. [ABSTRACT FROM AUTHOR]

Let { ξ( t) , t ∈ T} be a differentiable (in the mean-square sense) Gaussian random field with E ξ( t) ≡ 0, D ξ( t) ≡ 1, and continuous trajectories defined on the m-dimensional interval $ T \subset {\mathbb{R}^m} $. The paper is devoted to the problem of large excursions of the random field ξ. In particular, the asymptotic properties of the probability P = P{ −v( t) < ξ( t) < u( t) , t ∈ T}, when, for all t ∈ T, u( t) , v( t) ⩾ χ, χ → ∞, are investigated. The work is a continuation of Rudzkis research started in [R. Rudzkis, Probabilities of large excursions of empirical processes and fields, Sov. Math., Dokl., 45(1):226-228, 1992]. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then P = e + Qo(1), where Q is a certain constructive functional depending on u, v, T, and the matrix function R( t) = cov( ξ′( t) , ξ′( t)). [ABSTRACT FROM AUTHOR]

The paper deals with a mathematical model which describes how the collection of key phrases (and key words) evolves as a field of science develops. The experimental material is based on statistical observations on the sets of key phrases which have been assigned to papers in representative major journals in the field in question. Asymptotic properties of the model are considered, as well as estimators for the parameters of the model with particular emphasis on estimators that can indicate at what stage a collection of key phrases can be assumed as complete for the field in question at a given moment in time. [ABSTRACT FROM AUTHOR]