Duration Distribution of the Conjunction of Two Independent F Processes

Tan

2014

Statistics

In this paper we derive Piterbarg's max-discretisation theorem for two different grids considering centered stationary vector Gaussian processes. So far in the literature results in this direction have been derived for the joint distribution of the maximum of Gaussian processes over [0, T ] and over a grid R(δ 1 (T )) = {kδ 1 (T ) : k = 0, 1, · · · }. In this paper we extend the recent findings by considering additionally the maximum over another grid R(δ 2 (T )). We derive the joint limiting distribution of maximum of stationary Gaussian vector processes for different choices of such grids by letting T → ∞. As a by-product we find that the joint limiting distribution of the maximum over different grids, which we refer to as the Piterbarg distribution, is in the case of weakly dependent Gaussian processes a max-stable distribution.

Section: Introductionmentioning

confidence: 60%

Piterbarg's max-discretization theorem for stationary vector Gaussian processes observed on different grids

Tan

2014

Statistics

“…Typically, in applications such as the analysis of functional magnetic resonance imaging (fMRI) data, X i 's are assumed to be real-valued Gaussian random fields. Approximations of p T ,u are discussed for smooth Gaussian random fields in [22,5,10]; results for non-Gaussian random fields can be found in [6].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the probability of conjunctions of stationary Gaussian processes

Dȩbicki

Statistics & Probability Letters

et al. 2014

Let {X i (t), t ≥ 0}, 1 ≤ i ≤ n be independent centered stationary Gaussian processes with unit variance and almost surely continuous sample paths. For given positive constants u, T , define the set of conjunctions C [0,T ],u := {t ∈ [0, T ] : min 1≤i≤n X i (t) ≥ u}. Motivated by some applications in brain mapping and digital communication systems, we obtain exact asymptotic expansion of P C [0,T ],u = φ , as u → ∞.Moreover, we establish the Berman sojourn limit theorem for the random process {min 1≤i≤n X i (t), t ≥ 0} and derive the tail asymptotics of the supremum of each order statistics process.

“…We refer to, e.g., [2,7,32], for approximations of p T,u in the case of smooth Gaussian random fields. Results for non-Gaussian random fields and general stationary processes can be found in [3,12].…”

Section: Introductionmentioning

confidence: 99%

Extremes of vector-valued Gaussian processes: Exact asymptotics

Dȩbicki

Stochastic Processes and their Applications

et al. 2015

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.