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The purpose of this note is to show by constructing counterexamples that two conjectures of Móri and Székely for the Borel-Cantelli lemma are false.

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We obtain new lower and upper bounds for probabilities of unions of events. These bounds are sharp. They are stronger than earlier ones. General bounds may be applied in arbitrary measurable spaces. We have improved the method that has been introduced in previous papers. We derive new generalizations of the first and second parts of the Borel-Cantelli lemma.

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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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Moment inqualities and strong laws of large numbers are proved for random allocations of balls into boxes. Random broken lines and random step lines are constructed using partial sums of i.i.d. random variables that are modified by random allocations. Functional limit theorems for such random processes are obtained.

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