A New Class of Random Fields and Their Extreme Values

Simeon M. Berman
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Simeon M. Berman: Courant Institute of Mathematical Sciences

A chapter in Extreme Value Theory and Applications, 1994, pp 389-402 from Springer

Abstract: Abstract Let X be a random vector in R n , and let f(t), t ∈ R N , be a continuous function into R n for 1 ≤ N ≤ n. Let (x, y) be the usual inner product in R n , and let || x | | be the corresponding norm, for x,y ∈ R n . We define the real random field X(t) as X(t) = (X, f(t)), t ∈ R N . The basic hypothesis on the random vector is that it has a joint distribution which is invariant under all orthogonal transformations of R n , that is, X and U X have the same distribution for any real n × n orthogonal matrix U. The function f can be of a very general form, requiring continuous partial derivatives up to order 2. The marginal distributions of X(t) are assumed to be identical. A necessary and sufficient condition for this is that ||f(t)|| = constant in t. For u > 0 and T > 0, put L u = mes(t: t ∈ [0,T]N, X(t) > u), the sojourn “time” above the level u. Under the condition that the distribution of the random variable ||X|| is in the domain of attraction of the extreme value distribution exp(−e−x ), there is a function v(u) → ∞ for which we obtain the exact asymptotic form of the probability P(v(u)L u > x) for u → ∞, for any x > 0.

Keywords: Random Field; Random Vector; Marginal Distribution; Sojourn Time; Orthogonal Matrix (search for similar items in EconPapers)
Date: 1994
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DOI: 10.1007/978-1-4613-3638-9_23

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