Exact asymptotics and limit theorems for supremum of stationary<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>χ</mml:mi></mml:math>-processes over a random interval

Tan, Zhongquan; Hashorva, Enkelejd

doi:10.1016/j.spa.2013.03.009

Cited by 29 publications

(12 citation statements)

References 26 publications

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“…holds for any u large enough, with some positive constant C not depending on u. The last inequality is commonly referred to as the Piterbarg inequality; see e.g., Proposition 3.2 in [40] for the case of chi-processes.…”

Section: Preliminariesmentioning

confidence: 99%

Extremes and First Passage Times of Correlated Fractional Brownian Motions

Hashorva

2014

Stochastic Models

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Let {X i (t), t ≥ 0}, i = 1, 2 be two standard fractional Brownian motions being jointly Gaussian with constant cross-correlation. In this paper we derive the exact asymptotics of the joint survival functionA novel finding of this contribution is the exponential approximation of the joint conditional first passage times of X 1 , X 2 . As a by-product we obtain generalizations of the Borell-TIS inequality and the Piterbarg inequality for 2-dimensional Gaussian random fields.

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Section: Preliminariesmentioning

confidence: 99%

Extremes and First Passage Times of Correlated Fractional Brownian Motions

Hashorva

2014

Stochastic Models

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“…Hence, the claim for κ = 1 follows from (24) and (25) by choosing p > max(4/α + k, 2k). Case κ ∈ (1, ∞): Denote below by (Y (1) (t), Y (2) (t)) := r −1 1 (t)X 1 (0), .…”

Section: Appendixmentioning

confidence: 99%

Extremes and limit theorems for difference of chi-type processes

Albin

Hashorva

et al. 2016

ESAIM: PS

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Abstract. Let {ζ (κ) m,k (t), t ≥ 0}, κ > 0 be random processes defined as the differences of two independent stationary chi-type processes with m and k degrees of freedom. In this paper we derive the asymptotics of P sup t∈[0,T ] ζ (κ) m,k (t) > u , u → ∞ under some assumptions on the covariance structures of the underlying Gaussian processes. Further, we establish a Berman sojourn limit theorem and a Gumbel limit result.

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“…To our best knowledge, the study in the existing literature has only focused on χ-processes and fields indexed by intervals or hyper cubes, but not low-dimensional manifolds. See, for example, Albin et al [2], Bai [4], Hashorva and Ji [19], Ji et al [20], Konstantinides et al [22], Lindgren [23], Ling and Tan [24], Liu and Ji [25,26], Piterbarg [30,31], Tan and Hashorva [38,39], Tan and Wu [40]. Also it is worth mentioning that it is often assumed that X 1 , • • • , X r are independent copies of a Gaussian process or field X in the literature, while the cross-dependence among X 1 , • • • , X r is allowed under certain constraints in this work.…”

Section: Introductionmentioning

confidence: 99%

Extremes of locally stationary Gaussian and chi fields on manifolds

Qiao

2021

Stochastic Processes and their Applications

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Depending on a parameter h ∈ (0, 1], let {X h (t), t ∈ M h } be a class of centered Gaussian fields indexed by compact manifolds M h . For locally stationary Gaussian fields X h , we study the asymptotic excursion probabilities of X h on M h . Two cases are considered: (i) h is fixed and (ii) h → 0. These results are extended to obtain the limit behaviors of the extremes of locally stationary χ-fields on manifolds.

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Exact asymptotics and limit theorems for supremum of stationaryχ-processes over a random interval

Cited by 29 publications

References 26 publications

Extremes and First Passage Times of Correlated Fractional Brownian Motions

Extremes and First Passage Times of Correlated Fractional Brownian Motions

Extremes and limit theorems for difference of chi-type processes

Extremes of locally stationary Gaussian and chi fields on manifolds

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