This paper derives an exact asymptotic expression forwhere X(t) = (X 1 (t), . . . , X d (t)) ⊤ , t ≥ 0 is a correlated d-dimensional Brownian motion starting at the pointThe derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some cases to dimension-reduction of the considered problem. Complementary, we study asymptotic distribution of the conditional first passage time to U, which depends on the dimension-reduction phenomena.

Let {X H (t), t ≥ 0} be a fractional Brownian motion with Hurst index H ∈ (0, 1] and define a γ-In this paper we establish the exact tail asymptotic behaviour of M γ (T ) = sup t∈[0,T ] W γ (t) for any T ∈ (0, ∞].Furthermore, we derive the exact tail asymptotic behaviour of the supremum of certain non-homogeneous mean-zero Gaussian random fields.

Let {X i (t), t ≥ 0}, 1 ≤ i ≤ n be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of P ∃ t∈[0,T ] ∀ i=1,...,n X i (t) > u as u → ∞, for both locally stationary X i 's and X i 's with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants, that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell-TIS inequality, the Slepian lemma and the Pickands-Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes.2 KRZYSZTOF DȨ BICKI, ENKELEJD HASHORVA, LANPENG JI, AND KAMIL TABIŚ fields) including locally stationary Gaussian process and Gaussian process with a non-constant variance function. For a complete survey on related results we refer to [29,30].The main goal of this contribution is to derive exact asymptotics of (1) for large classes of non-stationary Gaussian processes X i 's, providing multidimensional counterparts of the seminal Pickands' and Piterbarg-Prishyaznyuk's results, respectively; see e.g., Theorem D2 and Theorem D3 in [29]. The proofs of our main results are based on an extension of the double-sum technique applied to the analysis of (1). Remarkably, the relation between (1) and (2) also implies the applicability of the double-sum method to non-Gaussian processes, as, e.g., the process {min 1≤i≤n X i (t), t ≥ 0}.Interestingly, in the obtained asymptotics, there appear multidimensional counterparts of the classical Pickands and Piterbarg constants (see Sections 2 and 3). We analyze properties of these new constants in Section 3.In the literature there are few results on extremes of non-smooth vector-valued Gaussian processes; see [4,15,22,34] and the references therein. In Section 5 we shall present some extensions (tailored for our use) of the Slepian lemma, the Borell-TIS inequality and the Piterbarg inequality for vector-valued Gaussian random fields. These results are of independent interest given their crucial role in the theory of Gaussian processes and random fields; see e.g., [1,8,26,29] and the references therein.The organization of the paper: Basic notation and some preliminary results are presented in Section 2. In Section 3 we analyze properties of vector-valued Pickands and Piterbarg constants. The main results of the paper, concerning the asymptotics of (1) for both locally stationary X i 's and X i 's with a non-constant generalized variance function, are displayed in Section 4. All the proofs are relegated to Section 5.

Let χ n (t) = ( n i=1 X 2 i (t)) 1/2 , t ≥ 0 be a chi-process with n degrees of freedom where X i 's are independent copies of some generic centered Gaussian process X. This paper derives the exact asymptotic behaviour ofwhere T is a given positive constant, and g(·) is some non-negative bounded measurable function. The case g(t) ≡ 0 has been investigated in numerous contributions by V.I. Piterbarg. Our novel asymptotic results, for both stationary and non-stationary X, are referred to as Piterbarg theorems for chi-processes with trend.

Let {X i (t), t ≥ 0}, 1 ≤ i ≤ n be independent copies of a stationary process {X(t), t ≥ 0}. For given positive constants u, T , define the set of rth conjunctions C r,T,u := {t ∈ [0, T ] : X r:n (t) > u} with X r:n (t) the rth largest order statistics of X i (t), t ≥ 0, 1 ≤ i ≤ n. In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions C r,T,u is not empty. Imposing the Albin's conditions on X, in this paper we obtain an exact asymptotic expansion of this probability as u tends to infinity. Furthermore, we establish the tail asymptotics of the supremum of the order statistics processes of skew-Gaussian processes and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes. As a by-product we derive a version of Li and Shao's normal comparison lemma for the minimum and the maximum of Gaussian random vectors.