On the Borel-Cantelli lemma

Necessary and sufficient conditions for P(A_n infinitely often) = , [0, 1], are obtained, where {A_n} is a sequence of events such that P(A _n ) = .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 200–207, 1990.     

On inequalities for probabilities of unions of events and the Borel-Cantelli lemma

New upper bound for probabilities of unions of events are found. Examples show their optimality. A generalization of the first part of the Borel-Cantelli lemma is obtained.     

The dynamical Borel-Cantelli lemma and the waiting time problems

We investigate the connection between the dynamical Borel-Cantelli and waiting time results. We prove that if a system has the dynamical Borel-Cantelli property, then the time needed to enter for the first time in a sequence of small balls scales as the inverse of the measure of the balls. Conversely if we know the waiting time behavior of a system we can prove that certain sequences of decreasing balls satisfies the Borel-Cantelli property. This allows to obtain Borel-Cantelli like results in systems like axiom A and generic interval exchanges.     

On the bramble-hilbert lemma

The aim of this paper is to give an improvement of a result by Bramble and Hilbert [2] concerning upper bounds for linear functionals on certain Sobolev spaces by showing that the constants in the estimate are independent of the bounded Lipschitz domain Ω.     

Borel-Cantelli sequences

A sequence {x _n } ₁ ^∞ in the unit interval [0, 1) = ?/? is called Borel-Cantelli, or BC, if for all non-increasing sequences of positive real numbers {a _n } with 48001013 1. \(\sum\nolimits_{i = 1}^\infty {} {a_i} = \infty \), the set

$\{ x \in [0,1)\left| {\left| {x - {x_n}} \right|} \right. < {a_n}{\rm{for infinitely many }}n \ge 1\} $

has full Lebesgue measure. (Speaking informally, BC sequences are sequences for which a natural converse to the Borel-Cantelli Theorem holds).

The notion of BC sequences is motivated by the monotone shrinking target property for dynamical systems, but our approach is from a geometric rather than dynamical perspective. A sufficient condition, a necessary condition and a necessary and sufficient condition for a sequence to be BC are established. A number of examples of BC sequences and sequences that are not BC are also presented.The property of a sequence to be BC is a delicate Diophantine property. For example, the orbits of a pseudo-Anosoff IET (interval exchange transformation) are BC, while the orbits of a “generic” IET are not.The notion of BC sequences is extended from [0, 1) to sequences in Ahlfors regular spaces.     

On the hadamard—perron lemma

A formal generalization of the Hadamard-Perron lemma to the nonhyperbolic case is considered.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 8 , Suzdal Conference—2, 2003.This revised version was published online in April 2005 with a corrected cover date.     

On a lemma of bowen

We are concerned with the sets of quasi generic points in finite symbolic space. We estimate the sizes of the sets by the Billingsley dimension defined by Gibbs measures. A dimension formula of such set is given, which generalizes Bowen's result. An application is given to the level sets of Birkhoff average.     

On a generalization of the Kalman-Yakubovic lemma

In this paper we generalize the Kalman-Yakubovic lemma to infinite dimensions—or, more precisely, to semigroups of operators over a Hilbert space. The proof differs substantially from the finite-dimensional version and is based on the Paley-Wiener-Helson-Lowdenslager factorization theorem.     

On the Hadamard lemma and the Lipschitz condition

We prove that the difference of values of a nonlinear Lipschitz continuous differential operator is always representable as the difference of values of some linear Lipschitz continuous differential operator with the same Lipschitz constant. The proof is based on the Hadamard lemma, provided that, in addition to the above requirements, the nonlinearity is continuously differentiable in spatial variables. In general case the proof is based on various criteria of the weak compactness and on various approximative statements obtained by the Steklov averaging technique.     

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In this paper, we will prove another Borel-Cantelli lemma for capacities induced by sublinear expectations.