Consider a dynamical systems $([0,1], T, \mu )$ which is exponentially mixing for $L^1$ against bounded variation. Given non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $\mu (B(x, r_k(x)) = m_k$. It prove

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

Let (An)n=1 be a sequence of sets in a probability space (X,B, μ) such that P∞ n=1 μ(An) =∞. The classical Borel-Cantelli lemma states that if the sets An are independent, then μ({x ∈ X : x ∈ An for infinitely many values of n}) = 1. We present analogous dynamical Borel-Cantelli lemmas for certain sequences of sets (An) inX (including nested balls) for a class of deterministic dynamical systems...

holds. This is the assertion of the Second Borel-Cantelli Lemma. If the assumption of independence is dropped, (1) fails in general. However, (1) remains true if independence is replaced by pairwise independence (Durrett (1991), Theorem (6.6) ch. 1) or by uniform mixing (Iosifescu and Theodorescu (1969) Lemma 1.1.2’). Rieders (1993) showed by an example that strong mixing without any further as...

Abstract We study the dynamical Borel–Cantelli lemma for recurrence sets in a measure-preserving system $(X, \mu , T)$ with compatible metric d . prove that under some regularity conditions, $\mu $ -measure of following set $$\begin{align*}R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\mathbb{N} \} \end{align*}$$ obeys zero–full law according to convergence or di...