Central limit theorems for extreme Sojourns of stationary Gaussian processes | Zendy

Berman Simeon M. | Zendy

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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

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.

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