185 research outputs found

    Sojourn times of Gaussian random fields

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    This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruitful, namely knowing the asymptotics of supremum of random processes and fields (in particular Gaussian) it is possible to establish the asymptotics of their sojourn times. We illustrate our findings considering i) two dimensional Gaussian random fields, ii) chi-process generated by stationary Gaussian processes and iii) stationary Gaussian queueing processes

    Sojourn times of Gaussian and related random fields

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    This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruitful, namely knowing the asymptotics of supremum of random processes and fields (in particular Gaussian) it is possible to establish the asymptotics of their sojourn times. We illustrate our findings considering i) two dimensional Gaussian random fields, ii) chi-process generated by stationary Gaussian processes and iii) stationary Gaussian queueing processes

    On the continuity of Pickands constants

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    Cluster Random Fields and Random-Shift Representations

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    This paper investigates random-shift representations of α\alpha-homogeneous shift-invariant classes of random fields (rf's) Kα[Z] K_{\alpha}[ Z], which were introduced in \cite{hashorva2021shiftinvariant}. Here Z(t),tT Z(t),t\in T is a stochastically continuous Rd\mathbb{R}^d-valued rf with T=Rl T=\mathbb{R}^l or T=ZlT=\mathbb{Z}^l. We show that that random-shift representations of interest are obtained by constructing cluster rf's, which play a crucial role in the study of extremes of stationary regularly varying rf's. An important implication of those representations is their close relationship with Rosi\'nski (or mixed moving maxima) representations of max-stable rf's. We show that for a given Kα[Z] K_{\alpha}[ Z] different cluster rf's can be constructed, which is useful for the derivation of new representations of extremal functional indices, Rosi\'nski representations of max-stable rf's as well as for random-shift representations of shift-invariant tail measures.Comment: 31 page

    Tail Measures and Regular Variation

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    A general framework for the study of regular variation is that of Polish star-shaped metric spaces, while recent developments in [1] have discussed regular variation in relation to a boundedness and weaker assumptions are imposed therein on the structure of Polish space. Along the lines of the latter approach, we discuss the regular variation of measures and processes on Polish spaces with respect to some given boundedness. We then focus on regular variation of cadlag processes on D(R^l, \R^d), which was recently studied for stationary cadlag processes on the real line in [2]. Tail measures introduced in [3] appear naturally as limiting measures of regularly varying random processes, and we show that this continues to hold true in our general setting. We derive several results for tail measures and their local tail/ spectral tail processes which are crucial for the investigation of regular variation of Borel measures.Comment: 33 page

    Piterbarg theorems for chi-processes with trend

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    Let χ n ( t ) = ( ∑ i = 1 n X i 2 ( t ) ) 1 / 2 , t ≥ 0 χn(t)=(i=1nXi2(t))1/2, t0\chi _{n}(t) = ({\sum }_{i=1}^{n} {X_{i}^{2}}(t))^{1/2},\ {t\ge 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 of 1 ℙ sup t ∈ [ 0 , T ] χ n ( t ) − g ( t ) > u as u → ∞ , P{supt[0,T](χn(t)g(t))>u}    as    u, \mathbb{P}\left\{\sup\limits_{t\in[0,T]} \left(\chi_{n}(t)- {g(t)} \right) > u\right\} \;\; \text{as} \;\; u \rightarrow \infty, where 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

    Pandemic-type failures in multivariate Brownian risk models

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    Modelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical dd-dimensional Brownian risk model (Brm), see [1]. From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. The main findings of this contribution concern the approximation of the probability that at least kk out of dd components of Brm fail simultaneously. We derive both sharp bounds and asymptotic approximations of the probability of interest for the finite and the infinite time horizon. Our results extend previous findings of [2,3].Comment: 28 page

    Multivariate max-stable processes and homogeneous functionals

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    Multivariate max-stable processes are important for both theoretical investigations and various statistical applications motivated by the fact that these are limiting processes, for instance of stationary multivariate regularly varying time series, [1]. In this contribution we explore the relation between homogeneous functionals and multivariate max-stable processes and discuss the connections between multivariate max-stable process and zonoid / max-zonoid equivalence. We illustrate our results considering Brown-Resnick and Smith processes
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