Some properties of conditionally independent random variables are studied. Conditional versions of generalized Borel-Cantelli lemma, generalized Kolmogorov’s inequality and generalized Hájek-Rényi inequality are proved. As applications, a conditional version of the strong law of large numbers for conditionally independent random variables and a conditional version of the Kolmogorov’s strong law...

The celebrated Borel-Cantelli lemma asserts that (A) If ~P(E,,) < 00, then P (lim sup Ek) =O; (B) If the events Es are independent and if xP(Ek) = m, then P(lim sup Eh) = 1. In intuitive language P(lim. sup &) is the probability that the events Eh occur “infinitely often” and will be denoted by P(Ek i.0.). This lemma is the basis of all theorems of the strong type in probability theory. Its app...

This note contains a Chover-type Law of the k -Iterated Logarithm for weighted sums of strong mixing sequences of random variables with a distribution in the domain of a stable law. We show that the upper part of the LIL is similar to other studies in the literature; conversely, the lower half is substantially different. In particular, we show that, due to the failure of the classical version o...

We prove that if a system has superpolynomial (faster than any power law) decay of correlations then the time τ r (x, x 0) needed for a typical point x to enter for the first time a ball B(x 0 , r) centered in x 0 , with small radius r scales as the local dimension at x 0 , i.e. lim r→0 log τ r (x, x 0) − log r = d µ (x 0). This result is obtained by proving a kind of dynamical Borel-Cantelli l...

This paper studies the asymptotic behavior for a class of delayed reaction-diffusion Hopfield neural networks driven by finite-dimensional Wiener processes. Some new sufficient conditions are established to guarantee the mean square exponential stability of this system by using Poincaré’s inequality and stochastic analysis technique. The proof of the almost surely exponential stability for this...