Extremes of threshold-dependent Gaussian processes

“…Extremes of locally stationary L p norm processes with trend. If X(t) is locally stationary, as in [6], we shall suppose that: (v) r(s, t) < 1, ∀ s, t ∈ [0, T ] and s = t.…”

Section: Extremes Of L P Norm Processes With Trendmentioning

confidence: 99%

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Extremes of L^p-norm of vector-valued Gaussian processes with trend

Bai

2018

Stochastics

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Let X(t) = (X 1 (t), . . . , X n (t)) be a Gaussian vector process and g(t) be a continuous function.The asymptotics of distribution of X(t) p , the L p norm for Gaussian finite-dimensional vector, have been investigated in numerous literatures. In this contribution we are concerned with the exact tail asymptotics of X(t) c p , c > 0, with trend g(t) over [0, T ]. Both scenarios that X(t) is locally stationary and nonstationary are considered. Important examples include n i=1 |X i (t)| + g(t) and chi-square processes with trend, i.e., n i=1 X 2 i (t) + g(t). These results are of interest in applications in engineering, insurance and statistics, etc.

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“…Finiteness of H α,a , P f α,a [0, ∞) and P f α,a (−∞, ∞) is guaranteed under some restrictions on f (·) which are satisfied in our setup; see [16,23,24]. We refer to, e.g., [24-26, 2, 4, 27-35] for properties of the above constants.…”

Section: Introductionmentioning

confidence: 99%

Extremes of vector-valued Gaussian processes with Trend

Bai

Dȩbicki

Liu

2018

Journal of Mathematical Analysis and Applications

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Let X(t) = (X 1 (t), . . . , X n (t)), t ∈ T ⊂ R be a centered vector-valued Gaussian process with independent components and continuous trajectories, and h(t) = (h 1 (t), . . . , h n (t)), t ∈ T be a vector-valued continuous function. We investigate the asymptotics of P sup t∈T min 1≤i≤n (X i (t) + h i (t)) > u as u → ∞. As an illustration to the derived results we analyze two important classes of X(t): with locallystationary structure and with varying variances of the coordinates, and calculate exact asymptotics of simultaneous ruin probability and ruin time in a Gaussian risk model. 1 2 LONG BAI, KRZYSZTOF DȨ BICKI, AND PENG LIUWe note that (2) can also be viewed as the probability that the conjunction set S T,u := {t ∈ [0, T ] : min 1≤i≤n (X i (t) + h i (t)) > u} is not empty in Gaussian conjunction problem, sincesee, e.g., [15,16] and references therein.The main results of this contribution extend recent findings of [16], where the exact asymptotics of (2) for h i ≡ 0, 1 ≤ i ≤ n was analyzed; see also [17] where X(t) is a multidimensional Brownian motion, h i (t) = c i t and T = ∞, and [18,19] for LDP-type results. It appears that the presence of the drift function substantially increases difficulty of the problem when comparing it with the analysis given for the driftless case in [16]. More specifically, as advocated in Section 2, it requires to deal with P sup.., n are families (with respect to u) of centered threshold-dependent Gaussian processes; see Theorem 2.1.In Section 3 we apply general results derived in Section 2 to two important families of Gaussian processes, i.e. i) to locally-stationary processes in the sense of Berman and ii) to processes with varying variance Var(X i (t)), t ∈ [0, T ]. Then, as an example to the derived theory, we analyze the probability of simultaneous ruin in Gaussian risk model. Complementary, we investigate the limit distribution of the simultaneous ruin timeOrganization of the rest of the paper: Section 2 is devoted to the main result of this contribution, concerning the extremes of the threshold-dependent centered Gaussian vector processes. In Section 3 we specify our result to locally-stationary vector-valued Gaussian processes with trend and non-stationary Gaussian vector-valued processes with trend. Detailed proofs of all the results are postponed to Section 4. Additionally, in Section 3 we analyze asymptotics of the simultaneous ruin probability. Main ResultsWe begin with observation that, for sufficiently large u,is a family of centered vector-valued threshold-dependent Gaussian processes. Since the above rearrangement appears to be useful for the technique of the proof that we use in order to get the exact asymptotics of (2), then in this section we focus on asymptotics of extremes of thresholddependent vector-valued Gaussian processes.More specifically, let X u (t) := (X u,1 (t), . . . , X u,n (t)), t ∈ E(u), with 0 ∈ E(u) = (x 1 (u), x 2 (u)), be a family of centered n-dimensional vector-valued Gaussian processes with continuous trajectories. Let σ 2 u,i (·) an...

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Extremes of threshold-dependent Gaussian processes

Cited by 10 publications

References 50 publications

Extremes of standard multifractional Brownian motion

Extremes of standard multifractional Brownian motion

Extremes of L^p-norm of vector-valued Gaussian processes with trend

Extremes of vector-valued Gaussian processes with Trend

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Extremes of threshold-dependent Gaussian processes

Cited by 10 publications

References 50 publications

Extremes of standard multifractional Brownian motion

Extremes of standard multifractional Brownian motion

Extremes of Lp-norm of vector-valued Gaussian processes with trend

Extremes of vector-valued Gaussian processes with Trend

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Extremes of L^p-norm of vector-valued Gaussian processes with trend