One of the oldest, most difficult, and most important problems in the theory of random processes has been the precise calculation of the probability
where X (t) is a random process. This problem is especially attractive for a Gaussian process. Even today, there is no explicit formula of the probability (1) in the general Gaussian situation, despite the fact that the set of finite dimensional distributions of a Gaussian process has a simple form. Because of this, there are multitudes of approximations and techniques of deriving approximations to (1), particularly when u is large. Gaussian processes appear in various fields (finance, hydrology, climatology, etc.) and finding the probability of attaining high level is particularly significant.
Extremes of Gaussian processes are well established in probability theory and mathematical statistics. Furthermore, besides the already mentioned problem, this theory addresses the...
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References and Further Reading
Adler R (2000) On excursion sets, tube formulas and maxima of random fields. Ann Appl Probab 20:1–74
Albeverio S, Piterbarg V (2006) Mathematical methods and concepts for the analysis of extreme events, In: Albeverio S, Jentsch V, Kantz H (eds) Extreme events in nature and society, Springer-Verlag, Berlin, pp 47–68
Leadbetter MR, Lindgren G, Rootzen H (1983) Extremes and related properties of random sequences and processes. Springer-Verlag, Berlin
Lifshits M (1995) Gaussian random functions, Kluwer, Dordrecht, The Netherlands
Pickands J III (1969) Upcrossing probabilities for stationary Gaussian processes. Trans Amer Math Soc 145:51–73
Piterbarg V (1996) Asymptotic methods in the theory of Gaussian processes and fields. (Translations of Mathematical Monographs), vol 148. AMS, Providence, RI
Piterbarg V, Fatalov V (1995) The Laplace method for probability measures in Banach Spaces, Russian Math Surv 50:1151–1239
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Stamatovic, S. (2011). Extremes of Gaussian Processes. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_245
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