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Journal articles on the topic 'Extremes of Gaussian processes'

Author: Grafiati

Published: 4 June 2021

Last updated: 20 February 2023

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1

Gong, K., and X. Z. Chen. "Estimating extremes of combined two Gaussian and non-Gaussian response processes." International Journal of Structural Stability and Dynamics 14, no. 03 (February 16, 2014): 1350076. http://dx.doi.org/10.1142/s0219455413500764.

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Assessment of structural performance under stochastic dynamic loadings requires estimation of the extremes of stochastic response components and the resultant responses as their linear and nonlinear combinations. This paper addresses the evaluations and combination rules for the extremes of scalar and vectorial resultant responses from two response components that may show non-Gaussian characteristics. The non-Gaussian response process is modeled as a translation process from an underlying Gaussian process. The mean crossing rates and extreme value distributions of resultant responses are calculated following the theory for vector-valued Gaussian processes. An extensive parameter study is conducted concerning the influence of statistical moments of non-Gaussian response components on the extremes of resultant responses. It is revealed that the existing combination rules developed for Gaussian processes are not applicable to the case of non-Gaussian process. New combination rules are suggested that permit predictions of the extremes of resultant responses directly from the extremes of response components.

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2

Dębicki, K., K. M. Kosiński, M. Mandjes, and T. Rolski. "Extremes of multidimensional Gaussian processes." Stochastic Processes and their Applications 120, no. 12 (December 2010): 2289–301. http://dx.doi.org/10.1016/j.spa.2010.08.010.

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3

Kabluchko, Zakhar. "Extremes of independent Gaussian processes." Extremes 14, no. 3 (April 6, 2010): 285–310. http://dx.doi.org/10.1007/s10687-010-0110-x.

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4

Piterbarg, V. I. "Large extremes of Gaussian chaos processes." Doklady Mathematics 93, no. 2 (March 2016): 145–47. http://dx.doi.org/10.1134/s1064562416020058.

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5

Kabluchko, Zakhar. "Extremes of space–time Gaussian processes." Stochastic Processes and their Applications 119, no. 11 (November 2009): 3962–80. http://dx.doi.org/10.1016/j.spa.2009.08.001.

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6

Dȩbicki, Krzysztof, Enkelejd Hashorva, and Longmin Wang. "Extremes of vector-valued Gaussian processes." Stochastic Processes and their Applications 130, no. 9 (September 2020): 5802–37. http://dx.doi.org/10.1016/j.spa.2020.04.008.

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7

Bai, Long, Krzysztof Dȩbicki, Enkelejd Hashorva, and Lanpeng Ji. "Extremes of threshold-dependent Gaussian processes." Science China Mathematics 61, no. 11 (September 5, 2018): 1971–2002. http://dx.doi.org/10.1007/s11425-017-9225-7.

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8

Toro, Gabriel R., and C. Allin Cornell. "Extremes of Gaussian Processes with Bimodal Spectra." Journal of Engineering Mechanics 112, no. 5 (May 1986): 465–84. http://dx.doi.org/10.1061/(asce)0733-9399(1986)112:5(465).

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9

Huesler, Juerg, Vladimir Piterbarg, and Yueming Zhang. "Extremes of Gaussian Processes with Random Variance." Electronic Journal of Probability 16 (2011): 1254–80. http://dx.doi.org/10.1214/ejp.v16-904.

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10

Bai, Long. "Extremes of Gaussian chaos processes with trend." Journal of Mathematical Analysis and Applications 473, no. 2 (May 2019): 1358–76. http://dx.doi.org/10.1016/j.jmaa.2019.01.026.

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11

Xu, Hui, and Mircea D. Grigoriu. "Finite dimensional models for extremes of Gaussian and non-Gaussian processes." Probabilistic Engineering Mechanics 68 (April 2022): 103199. http://dx.doi.org/10.1016/j.probengmech.2022.103199.

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12

Leira, Bernt J. "Extremes of Gaussian and non-Gaussian vector processes: a geometric approach." Structural Safety 25, no. 4 (October 2003): 401–22. http://dx.doi.org/10.1016/s0167-4730(03)00017-1.

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13

Dȩbicki, Krzysztof, Enkelejd Hashorva, and Peng Liu. "Extremes ofγ-reflected Gaussian processes with stationary increments." ESAIM: Probability and Statistics 21 (2017): 495–535. http://dx.doi.org/10.1051/ps/2017019.

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14

Dȩbicki, Krzysztof, Enkelejd Hashorva, Lanpeng Ji, and Kamil Tabiś. "Extremes of vector-valued Gaussian processes: Exact asymptotics." Stochastic Processes and their Applications 125, no. 11 (November 2015): 4039–65. http://dx.doi.org/10.1016/j.spa.2015.05.015.

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15

Dieker, A. B. "Extremes of Gaussian processes over an infinite horizon." Stochastic Processes and their Applications 115, no. 2 (February 2005): 207–48. http://dx.doi.org/10.1016/j.spa.2004.09.005.

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16

Hüsler, J., and V. Piterbarg. "Extremes of a certain class of Gaussian processes." Stochastic Processes and their Applications 83, no. 2 (October 1999): 257–71. http://dx.doi.org/10.1016/s0304-4149(99)00041-1.

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17

Zhao, Chunming. "Extremes of order statistics of stationary Gaussian processes." Probability and Mathematical Statistics 38, no. 1 (July 30, 2018): 61–75. http://dx.doi.org/10.19195/0208-4147.38.1.4.

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18

Bai, Long, Krzysztof Dȩbicki, and Peng Liu. "Extremes of vector-valued Gaussian processes with Trend." Journal of Mathematical Analysis and Applications 465, no. 1 (September 2018): 47–74. http://dx.doi.org/10.1016/j.jmaa.2018.04.069.

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19

Piterbarg*, Vladimir I. "Discrete and Continuous Time Extremes of Gaussian Processes." Extremes 7, no. 2 (June 2004): 161–77. http://dx.doi.org/10.1007/s10687-005-6198-8.

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20

Albin, Patrik. "On extreme value theory for group stationary Gaussian processes." ESAIM: Probability and Statistics 22 (2018): 1–18. http://dx.doi.org/10.1051/ps/2018002.

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We study extreme value theory of right stationary Gaussian processes with parameters in open subsets with compact closure of (not necessarily Abelian) locally compact topological groups. Even when specialized to Euclidian space our result extend results on extremes of stationary Gaussian processes and fields in the literature by means of requiring weaker technical conditions as well as by means of the fact that group stationary processes need not be stationary in the usual sense (that is, with respect to addition as group operation).

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21

Piterbarg, Vladimir, Goran Popivoda, and Sinisa Stamatovic. "Extremes of Gaussian processes with a smooth random trend." Filomat 31, no. 8 (2017): 2267–79. http://dx.doi.org/10.2298/fil1708267p.

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Let ?(t), t ? R, be a Gaussian zero mean stationary process, and ?(t) another random process, smooth enough, being independent of ?(t). We will consider the process ?(t) + ?(t) such that conditioned on ?(t) it is a Gaussian process. We want to establish an asymptotic exact result for P (t?[o,T] sup (?(t) + ?(t)) > u), as u ? ?, where T > 0.

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22

Dȩbicki, Krzysztof, and Kamil Tabiś. "Extremes of the time-average of stationary Gaussian processes." Stochastic Processes and their Applications 121, no. 9 (September 2011): 2049–63. http://dx.doi.org/10.1016/j.spa.2011.05.005.

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23

Hüsler, Jürg, Vladimir Piterbarg, and Ekaterina Rumyantseva. "Extremes of Gaussian processes with a smooth random variance." Stochastic Processes and their Applications 121, no. 11 (November 2011): 2592–605. http://dx.doi.org/10.1016/j.spa.2011.06.006.

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24

Hüsler, J. "Extremes of Gaussian processes, on results of Piterbarg and Seleznjev." Statistics & Probability Letters 44, no. 3 (September 1999): 251–58. http://dx.doi.org/10.1016/s0167-7152(99)00016-4.

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25

Leira, Bernt J. "Multivariate distributions of maxima and extremes for Gaussian vector-processes." Structural Safety 14, no. 4 (July 1994): 247–65. http://dx.doi.org/10.1016/0167-4730(94)90014-0.

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26

Tan, Zhongquan. "Limit laws on extremes of nonhomogeneous Gaussian random fields." Journal of Applied Probability 54, no. 3 (September 2017): 811–32. http://dx.doi.org/10.1017/jpr.2017.36.

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Abstract In this paper, by using the tail asymptotics derived by Dębicki et al. (2016), we prove the Gumbel limit laws for the maximum of a class of nonhomogeneous Gaussian random fields. As an application of the main results, we derive the Gumbel limit law for Shepp statistics of fractional Brownian motion and Gaussian integrated processes.

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27

Dieker, A. B., and B. Yakir. "On asymptotic constants in the theory of extremes for Gaussian processes." Bernoulli 20, no. 3 (August 2014): 1600–1619. http://dx.doi.org/10.3150/13-bej534.

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28

Anshin, A. B. "On the Probability of Simultaneous Extremes of Two Gaussian Nonstationary Processes." Theory of Probability & Its Applications 50, no. 3 (January 2006): 353–66. http://dx.doi.org/10.1137/s0040585x97981809.

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29

Tan, Zhongquan, and Yuebao Wang. "Extremes Values of Discrete and Continuous Time Strongly Dependent Gaussian Processes." Communications in Statistics - Theory and Methods 42, no. 13 (July 3, 2013): 2451–63. http://dx.doi.org/10.1080/03610926.2011.611322.

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30

Bai, Long. "Extremes of Lp-norm of vector-valued Gaussian processes with trend." Stochastics 90, no. 8 (August 13, 2018): 1111–44. http://dx.doi.org/10.1080/17442508.2018.1499101.

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31

Zhdanov, A. I., and V. I. Piterbarg. "High Extremes of Gaussian Chaos Processes: A Discrete Time Approximation Approach." Theory of Probability & Its Applications 63, no. 1 (January 2018): 1–21. http://dx.doi.org/10.1137/s0040585x97t988885.

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32

van der Hofstad, Remco, and Harsha Honnappa. "Large deviations of bivariate Gaussian extrema." Queueing Systems 93, no. 3-4 (October 15, 2019): 333–49. http://dx.doi.org/10.1007/s11134-019-09632-z.

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Abstract We establish sharp tail asymptotics for componentwise extreme values of bivariate Gaussian random vectors with arbitrary correlation between the components. We consider two scaling regimes for the tail event in which we demonstrate the existence of a restricted large deviations principle and identify the unique rate function associated with these asymptotics. Our results identify when the maxima of both coordinates are typically attained by two different versus the same index, and how this depends on the correlation between the coordinates of the bivariate Gaussian random vectors. Our results complement a growing body of work on the extremes of Gaussian processes. The results are also relevant for steady-state performance and simulation analysis of networks of infinite server queues.

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33

Dawley, Shawn, Yong Zhang, Xiaoting Liu, Peng Jiang, Geoffrey Tick, HongGuang Sun, Chunmiao Zheng, and Li Chen. "Statistical Analysis of Extreme Events in Precipitation, Stream Discharge, and Groundwater Head Fluctuation: Distribution, Memory, and Correlation." Water 11, no. 4 (April 5, 2019): 707. http://dx.doi.org/10.3390/w11040707.

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Hydrological extremes in the water cycle can significantly affect surface water engineering design, and represents the high-impact response of surface water and groundwater systems to climate change. Statistical analysis of these extreme events provides a convenient way to interpret the nature of, and interaction between, components of the water cycle. This study applies three probability density functions (PDFs), Gumbel, stable, and stretched Gaussian distributions, to capture the distribution of extremes and the full-time series of storm properties (storm duration, intensity, total precipitation, and inter-storm period), stream discharge, lake stage, and groundwater head values observed in the Lake Tuscaloosa watershed, Alabama, USA. To quantify the potentially non-stationary statistics of hydrological extremes, the time-scale local Hurst exponent (TSLHE) was also calculated for the time series data recording both the surface and subsurface hydrological processes. First, results showed that storm duration was most closely related to groundwater recharge compared to the other storm properties, while intensity also had a close relationship with recharge. These relationships were likely due to the effects of oversaturation and overland flow in extreme total precipitation storms. Second, the surface water and groundwater series were persistent according to the TSLHE values, because they were relatively slow evolving systems, while storm properties were anti-persistent since they were rapidly evolving in time. Third, the stretched Gaussian distribution was the most effective PDF to capture the distribution of surface and subsurface hydrological extremes, since this distribution can capture the broad transition from a Gaussian distribution to a power-law one.

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34

Zhdanov, A. I. "On Probability of High Extremes for Product of Two Gaussian Stationary Processes." Theory of Probability & Its Applications 60, no. 3 (January 2016): 520–27. http://dx.doi.org/10.1137/s0040585x97t987818.

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35

Piterbarg, Vladimir, Goran Popivoda, and Siniša Stamatović. "Extremes of Gaussian processes with smooth random expectation and smooth random variance." Lithuanian Mathematical Journal 57, no. 1 (January 2017): 128–41. http://dx.doi.org/10.1007/s10986-017-9347-2.

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36

Huesler, Juerg, Anna Ladneva, and Vladimir Piterbarg. "On Clusters of High Extremes of Gaussian Stationary Processes with $\varepsilon$-Separation." Electronic Journal of Probability 15 (2010): 1825–62. http://dx.doi.org/10.1214/ejp.v15-828.

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37

Bai, Long. "Extremes of α(t)-locally stationary Gaussian processes with non-constant variances." Journal of Mathematical Analysis and Applications 446, no. 1 (February 2017): 248–63. http://dx.doi.org/10.1016/j.jmaa.2016.08.056.

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38

Davison, A. C., and M. M. Gholamrezaee. "Geostatistics of extremes." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2138 (October 12, 2011): 581–608. http://dx.doi.org/10.1098/rspa.2011.0412.

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We describe a prototype approach to flexible modelling for maxima observed at sites in a spatial domain, based on fitting of max-stable processes derived from underlying Gaussian random fields. The models we propose have generalized extreme-value marginal distributions throughout the spatial domain, consistent with statistical theory for maxima in simpler cases, and can incorporate both geostatistical correlation functions and random set components. Parameter estimation and fitting are performed through composite likelihood inference applied to observations from pairs of sites, with occurrence times of maxima taken into account if desired, and competing models are compared using appropriate information criteria. Diagnostics for lack of model fit are based on maxima from groups of sites. The approach is illustrated using annual maximum temperatures in Switzerland, with risk analysis proposed using simulations from the fitted max-stable model. Drawbacks and possible developments of the approach are discussed.

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39

Rusakov, Oleg V., and Roman A. Ragozin. "On extremes of PSI-processes and gaussian limits of their normalized independent identical distributed sums." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 9, no. 2 (2022): 269–77. http://dx.doi.org/10.21638/spbu01.2022.208.

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We define PSI-process — Poisson Stochastic Index process, as a continuous time random process which is obtained by a manner of a randomization for the discrete time of a random sequence. We consider the case when a double stochastic Poisson process generates this randomization, i. e. such Poisson process has a random intensity. Under condition of existence of the second moment the stationary PSI-processes possess a covariance which coincides with the Laplace transform of the random intensity. In our paper we derive distributions of extremes for a one PSI-process, and these extremes are expressed in terms of Laplace transform of the random intensity. The second task that we solve is a convergence of the maximum of Gaussian limit for normalized sums of i. i. d. stationary PSI-processes. We obtain necessary and sufficient conditions for the intensity under which, after proper centering and normalization, this Gaussian limit converges in distribution to the double Exponential Law. For solution this task we essentially base on the monograph: M.R.Leadbetter, Georg Lindgren, Holder Rootzen (1986) “Extremes and Relative Properties of Random Sequences and Processes”, end essentially use the Tauberian theorem in W. Feller form.

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40

Piterbarg, Vladimir I., and Alexander Zhdanov. "On probability of high extremes for product of two independent Gaussian stationary processes." Extremes 18, no. 1 (August 31, 2014): 99–108. http://dx.doi.org/10.1007/s10687-014-0205-x.

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41

Albin, J. M. P. "Extremes and crossings for differentiable stationary processes with application to Gaussian processes in Rm and Hilbert space." Stochastic Processes and their Applications 42, no. 1 (August 1992): 119–47. http://dx.doi.org/10.1016/0304-4149(92)90030-t.

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42

Garfinkel, Chaim I., and Nili Harnik. "The Non-Gaussianity and Spatial Asymmetry of Temperature Extremes Relative to the Storm Track: The Role of Horizontal Advection." Journal of Climate 30, no. 2 (January 2017): 445–64. http://dx.doi.org/10.1175/jcli-d-15-0806.1.

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The distribution of near-surface and tropospheric temperature variability in midlatitudes is distinguishable from a Gaussian in meteorological reanalysis data; consistent with this, warm extremes occur preferentially poleward of the location of cold extremes. To understand the factors that drive this non-Gaussianity, a dry general circulation model and a simple model of Lagrangian temperature advection are used to investigate the connections between dynamical processes and the occurrence of extreme temperature events near the surface. The non-Gaussianity evident in reanalysis data is evident in the dry model experiments, and the location of extremes is influenced by the location of the jet stream and storm track. The cause of this in the model can be traced back to the synoptic evolution within the storm track leading up to cold and warm extreme events: negative temperature extremes occur when an equatorward propagating high–low couplet (high to the west) strongly advects isotherms equatorward over a large meridional fetch over more than two days. Positive temperature anomalies occur when a poleward propagating low–high couplet (low to the west) advects isotherms poleward over a large meridional fetch over more than two days. The magnitude of the extremes is enhanced by the meridional movement of the systems. Overall, horizontal temperature advection by storm track systems can account for the warm/cold asymmetry in the latitudinal distribution of the temperature extremes.

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43

Rusakov, O. V., and R. A. Ragozin. "Extremes of PSI-Processes and Gaussian Limits of Their Normalized Independent Identically Distributed Sums." Vestnik St. Petersburg University, Mathematics 55, no. 2 (June 2022): 186–91. http://dx.doi.org/10.1134/s106345412202011x.

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44

Kratz, Marie F., and JoséR León. "Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: Crossings and extremes." Stochastic Processes and their Applications 66, no. 2 (March 1997): 237–52. http://dx.doi.org/10.1016/s0304-4149(96)00122-6.

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45

Hashorva, Enkelejd, and Lanpeng Ji. "Approximation of Passage Times of γ-Reflected Processes with FBM Input." Journal of Applied Probability 51, no. 3 (September 2014): 713–26. http://dx.doi.org/10.1239/jap/1409932669.

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Define a γ-reflected processWγ(t) =YH(t) - γinfs∈[0,t]YH(s),t≥ 0, with input process {YH(t),t≥ 0}, which is a fractional Brownian motion with Hurst indexH∈ (0, 1) and a negative linear trend. In risk theoryRγ(u) =u-Wγ(t),t≥ 0, is referred to as the risk process with tax payments of a loss-carry-forward type. For various risk processes, numerous results are known for the approximation of the first and last passage times to 0 (ruin times) when the initial reserveugoes to ∞. In this paper we show that, for the γ-reflected process, the conditional (standardized) first and last passage times are jointly asymptotically Gaussian and completely dependent. An important contribution of this paper is that it links ruin problems with extremes of nonhomogeneous Gaussian random fields defined byYH, which we also investigate.

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46

Low, Y. M. "Extreme value analysis of bimodal Gaussian processes." Journal of Sound and Vibration 330, no. 14 (July 2011): 3458–72. http://dx.doi.org/10.1016/j.jsv.2011.01.033.

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47

Hashorva, Enkelejd, and Lanpeng Ji. "Approximation of Passage Times of γ-Reflected Processes with FBM Input." Journal of Applied Probability 51, no. 03 (September 2014): 713–26. http://dx.doi.org/10.1017/s0021900200011621.

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Define a γ-reflected process W γ(t) = Y H (t) - γinf s∈[0,t] Y H (s), t ≥ 0, with input process {Y H (t), t ≥ 0}, which is a fractional Brownian motion with Hurst index H ∈ (0, 1) and a negative linear trend. In risk theory R γ(u) = u - W γ(t), t ≥ 0, is referred to as the risk process with tax payments of a loss-carry-forward type. For various risk processes, numerous results are known for the approximation of the first and last passage times to 0 (ruin times) when the initial reserve u goes to ∞. In this paper we show that, for the γ-reflected process, the conditional (standardized) first and last passage times are jointly asymptotically Gaussian and completely dependent. An important contribution of this paper is that it links ruin problems with extremes of nonhomogeneous Gaussian random fields defined by Y H , which we also investigate.

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48

Konakov, Valentin, Vladimir Panov, and Vladimir Piterbarg. "Extremes of a class of non-stationary Gaussian processes and maximal deviation of projection density estimates." Extremes 24, no. 3 (February 10, 2021): 617–51. http://dx.doi.org/10.1007/s10687-020-00402-2.

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49

Konstant, D. G., and V. I. Piterbarg. "Extreme values of the cyclostationary Gaussian random process." Journal of Applied Probability 30, no. 1 (March 1993): 82–97. http://dx.doi.org/10.2307/3214623.

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In this paper the class of cyclostationary Gaussian random processes is studied. Basic asymptotics are given for the class of Gaussian processes that are centered and differentiable in mean square. Then, under certain conditions on the non-degeneration of the centered cyclostationary Gaussian process with integrable covariance functions, the Gnedenko-type limit formula is established for and all x > 0.

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50

Konstant, D. G., and V. I. Piterbarg. "Extreme values of the cyclostationary Gaussian random process." Journal of Applied Probability 30, no. 01 (March 1993): 82–97. http://dx.doi.org/10.1017/s0021900200044016.

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In this paper the class of cyclostationary Gaussian random processes is studied. Basic asymptotics are given for the class of Gaussian processes that are centered and differentiable in mean square. Then, under certain conditions on the non-degeneration of the centered cyclostationary Gaussian process with integrable covariance functions, the Gnedenko-type limit formula is established for and all x > 0.

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