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Andrei N. Frolov

Publications and source records attributed to Andrei N. Frolov.

7 recordsLinked to original sources
arXiv2021

On strong forms of the Borel--Cantelli lemma and intermittent interval maps

We derive new variants of the quantitative Borel--Cantelli lemma and apply them to analysis of statistical properties for some dynamical systems. We consider intermittent maps of $(0,1]$ which have absolutely continuous invariant probability measures. In particular, we prove that every sequence of intervals with left endpoints uniformly separated from zero is the strong Borel--Cantelli sequence with respect to such map and invariant measure.

math.PR↗
arXiv2020

On upper and lower bounds for probabilities of combinations of events

We derive new upper and lower bounds for probabilities that $r$ or at least $r$ from $n$ events occur. These bounds can turn to equalities. The method is discussed as well. It works for measurable space and measures with sign, too. We also discuss variants of the results for conditional probability of above events given $σ$-field. Taking expectations from both parts of inequalities for conditional probabilities can yield better bounds for unconditional ones.

math.PR↗
arXiv2019

On large deviations for combinatorial sums

We investigate asymptotic behaviour of probabilities of large deviations for normalized combinatorial sums. We find a zone in which these probabilities are equivalent to the tail of the standard normal law. Our conditions are similar to the classical Bernstein condition. The range of the zone of the normal convergence can be of power order.

math.PR↗
arXiv2014

On lower and upper bounds for probabilities of unions and the Borel--Cantelli lemma

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 maybe 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 part of the Borel--Cantelli lemma.

math.PR↗
arXiv2014

Bounds of the remainder in a combinatorial central limit theorem

We derive new bounds of the remainder in a combinatorial central limit theorem without assumptions on independence and existence of moments of summands. For independent random variables our theorems imply Esseen and Berry-Esseen type inequalities, some other new bounds and a combinatorial central limit theorem in the case of infinite variations.

math.PR↗
arXiv2012

Small deviations of iterated processes in space of trajectories

We derive logarithmic asymptotics of probabilities of small deviations for iterated processes in the space of trajectories. We find conditions under which these asymptotics coincide with those of processes generating iterated processes. When these conditions fail the asymptotics are quite different.

math.PR↗