Related papers: Multiple Borel Cantelli Lemma in dynamics and Mult…

Multiple Borel-Cantelli Lemma is a criterion that characterizes the occurrence of multiple rare events on the same time scale. We generalize the multiple Borel-Cantelli Lemma in dynamics established by Dolgopyat, Fayad and Liu [J. Mod. Dyn.…

The most versatile version of the classical divergence Borel-Cantelli lemma shows that for any divergent sequence of events $E_n$ in a probability space satisfying a quasi-independence condition, its corresponding limsup set $E_\infty$ has…

Consider a mixing dynamical systems $([0,1], T, \mu)$, for instance a piecewise expanding interval map with a Gibbs measure $\mu$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $\mu (B(x,…

We quantify the elementary Borel-Cantelli Lemma by higher moments of the overlap count statistic in terms of the weighted summability of the probabilities. Applications include mean deviation frequencies in the Strong Law and the Law of the…

Let $(X,T,\mu,d)$ be a metric measure-preserving system for which $3$-fold correlations decay exponentially for Lipschitz continuous observables. Suppose that $(M_k)$ is a sequence satisfying some weak decay conditions and suppose there…

We give a version of the Borel-Cantelli lemma. As an application, we prove an almost sure local central limit theorem. As another application, we prove a dynamical Borel-Cantelli lemma for systems with sufficiently fast decay of…

Let $T: X\mapsto X$ be a deterministic dynamical system preserving a probability measure $\mu$. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsets $A_n\subset X$ and $\mu$-almost every point $x\in X$ the…

Let $ ([0,1]^d,T,\mu) $ be a measure-preserving dynamical system so that the correlations decay exponentially for H\"older continuous functions. Suppose that $ \mu $ is absolutely continuous with a density function $ h\in L^q(\mathcal L^d)…

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…

We establish dynamical Borel-Cantelli lemmas for nested balls and rectangles centered at generic points in the setting of geometric Lorenz maps. We also establish extreme value statistics for observations maximized at generic points for…

We establish functional limit theorems for ergodic sums of observables with power singularities for expanding circle maps. In the regime where the observables have infinite variance, we show that when rescaled by $N^{1/s}(\ln N)^\alpha$,…

The object of this paper is twofold. From one side we study the dichotomy, in terms of the Extremal Index of the possible Extreme Value Laws, when the rare events are centred around periodic or non periodic points. Then we build a general…

In this paper, by establishing a Borel-Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random…

For a sequence of independent events $E_n$ the sum of the associated zero-one random variables $1_{E_n}$ is almost surely finite or almost surely infinite according as the sum of the probabilities converges or diverges. In this paper the…

In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero-one…

In the general context of computable metric spaces and computable measures we prove a kind of constructive Borel-Cantelli lemma: given a sequence (constructive in some way) of sets $A_{i}$ with effectively summable measures, there are…

Let (X k) be a strictly stationary sequence of random variables with values in some Polish space E and common marginal $\mu$, and (A k) k>0 be a sequence of Borel sets in E. In this paper, we give some conditions on (X k) and (A k) under…

Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate…

We prove dynamical Borel Canteli Lemmas for discrete time homogenous flows hitting a sequence of shrinking targets in a hyperbolic manifold. These results apply to both diagonalizable and unipotent flows, and any family of measurable…

In this paper, we investigate annealed and quenched limit theorems for random expanding dynamical systems. Making use of functional analytic techniques and more probabilistic arguments with martingales, we prove annealed versions of a…