We discuss a method which yields new bounds for probabilities of unions of events. These bounds are stronger than the Chung–Erdős inequality and its generalizations. We derive new generalizations of the second part of the Borel–Cantelli lemma. Earlier generalizations are special cases.

We obtain converses to the one-sided strong laws for large increment of sums of independent identically distributed (i.i.d.) random variables with a distribution from the domain of attraction of a normal law or a completely asymmetric stable law with index [alpha][set membership, variant](1,2).

We investigate the almost sure behaviour of the length of the longest increasing run in multivariate case.

We obtain necessary and sufficient conditions for one-sided strong laws of large numbers and laws of the iterated logarithm for large increments of sums of i.i.d. random variables. Our results are generalizations of Csörgo-Révész results on strong approximation laws.