The goal of this paper is an exhaustive investigation of the link between the tail measure of a regularly varying time series and its spectral tail process, independently introduced in Owada and Samorodnitsky (2012) and Basrak and Segers (2009). Our main result is to prove in an abstract framework that there is a one to one correspondance between these two objets, and given one of them to show that it is always possible to build a time series of which it will be the tail measure or the spectral tail process. For non negative time series, we recover results explicitly or implicitly known in the theory of maxstable processes. Tail measures on a metric space.2.1. Framework. The mathematical setting is the following. Let pE, Eq be a measurable cone, that is a measurable space together with a multiplication by positive scalars pu, xq P p0, 8qˆE Þ Ñ ux P E , which is measurable with respect to the product σ-field Bp0, 8q b E{E and satisfies 1x " x , upvpxqq " puvqx , u, v ą 0 , x P E .We assume that the cone admits a zero element 0 E P E such that u0 E " 0 E for all u ą 0 and that it is endowed with a pseudonorm, i.e. a measurable function }¨} E : E Þ Ñ r0, 8q such that }ux} E " u}x} E for all u ą 0, x P E and }x} E " 0 implies x " 0 E . The triangle inequality is not required.The space E Z of E-valued sequences is endowed with the cylinder σ-algebra F " E bZ and a generic sequence is denoted x " px h q hPZ . The sequence identically equal to 0 E is denoted by imsart-aap ver.

Multivariate Mills ratio, Gaussian random sequences, tail asymptotics, quadratic programming,

Let {X n , n 1} be a sequence of centered Gaussian random vectors in R d , d 2. In this paper we obtain asymptotic expansions (n → ∞) of the tail probability P{X n > t n } with t n ∈ R d a threshold with at least one component tending to infinity. Upper and lower bounds for this tail probability and asymptotics of discrete boundary crossings of Brownian Bridge are further discussed.

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.

In this paper we discuss the asymptotic behaviour of random contractions X = RS, where R, with distribution function F , is a positive random variable independent of S ∈ (0, 1).Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of X assuming that F is in the max-domain of attraction of an extreme value distribution and the distribution function of S satisfies a regular variation property.We apply our result to derive the asymptotics of the probability of ruin for a particular discretetime risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.

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.