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Abstract  

We establish some large increment results for partial sum processes of a dependent stationary Gaussian sequence via estimating upper bounds of large deviation probabilities on suprema of the Gaussian sequence.

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Abstract  

We establish strong limsup theorems related to the law of the iterated logarithm (LIL) for finite dimensional Gaussian random fields by using the second Borel-Cantelli lemma.

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Summary An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.

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SENETA, E., Regularly varying functions , Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 56 # 12189. Russian edition: Pravil'no menyayushchiesya funktsii , Nauka, Moskva, 1985. MR 86m :26001

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