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Central limit theorems for extreme Sojourns of stationary Gaussian processes
Author(s) -
Berman Simeon M.
Publication year - 1991
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160440807
Subject(s) - mathematics , limit (mathematics) , gaussian , covariance , covariance function , combinatorics , distribution (mathematics) , central limit theorem , gaussian process , stationary process , function (biology) , mathematical analysis , statistics , physics , quantum mechanics , evolutionary biology , biology
Abstract Let X ( t ), t ≧ 0, be a real stationary Gaussian process, and, for u > 0 and t >0, let L t ( u ) be the time spent by X ( s ), 0 ≦ s ≦ t , above the level u . Here u is taken to be a function u ( t ) of t , and L t is defined as L t ( u ( t )). It is shown that the distribution of ( L t – EL t )/(Var L t ) 1/2 converges, for t → ∞ and u ( t ) → ∞, to a standard normal distribution under various conditions relating the growth of u ( t ) to the decay of the covariance and other functions associated with it.