Borel-Cantelli Lemma Research Articles

Exponential stability of the split-step θ-method for neutral stochastic delay differential equations with jumps

The exponential mean-square stability of the split-step θ-method for neutral stochastic delay differential equations (NSDDEs) with jumps is considered. New conditions for jumps are proposed to ensure the exponential mean-square stability of the trivial solution. If the drift coefficient satisfies the linear growth condition, it is shown that the split-step θ-method can reproduce the exponential mean-square stability of the trivial solution for the constrained stepsize. Then by applying the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure exponential stability of both the trivial solution and the numerical solution can be obtained. Since split-step θ-method covers Euler–Maruyama (EM) method and split-step backward Euler (SSBE) method, the conclusions are valid for these two methods. Moreover, they can adapt to the NSDDEs and the SDDEs with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.

Dynamical Borel–Cantelli lemmas and rates of growth of Birkhoff sums of non-integrable observables on chaotic dynamical systems

We consider implications of dynamical Borel–Cantelli lemmas for rates of growth of Birkhoff sums of non-integrable observables , k > 0, on ergodic dynamical systems where . Some general results are given as well as some more concrete examples involving non-uniformly expanding maps, intermittent type maps as well as uniformly hyperbolic systems.

Stability analysis of impulsive stochastic Cohen–Grossberg neural networks driven by G-Brownian motion

ABSTRACTThis paper is devoted to study the stability of a class of impulsive stochastic Cohen–Grossberg neural networks driven by G-Brownian motion. By means of G-Lyapunov function, Borel–Cantelli lemma and inequality technique, a series of sufficient conditions on pth moment stability and quasi-sure stability with respect to a general decay function are established. A concrete example is given to illustrate the obtained results.

Kernel classification with missing data and the choice of smoothing parameters

Methods are proposed for selecting smoothing parameters of kernel classifiers in the presence of missing covariates. Here the missing covariates can appear in both the data and in the unclassified observation that has to be classified. The proposed methods are quite straightforward to implement. Exponential performance bounds will be derived for the resulting classifiers. Such bounds, in conjunction with the Borel–Cantelli lemma, provide various strong consistency results. Several numerical examples are presented to illustrate the effectiveness of the proposed procedures.

Parameter estimation for partially observed nonlinear stochastic system

This paper is concerned with the problem of parameter estimation for a partially observed linear stochastic system. The state estimator is obtained by using the continuous-time Kalman linear filtering theory. The likelihood function is given based on the innovation theorem and Girsanov theorem, the parameter estimator and error of estimation are derived. The strong consistency of the parameter estimator and the asymptotic normality of the error of estimation are proved by applying ergodic theorem, maximal inequality for martingale, Borel-Cantelli lemma and the central limit theorem for stochastic integrals.

Existence of Random Attractor for Stochastic Fractional Long-Short Wave Equations with Periodic Boundary Condition

We consider the asymptotic behaviors of stochastic fractional long-short equations driven by a random force. Under a priori estimates in the sense of expectation, using Galerkin approximation by the stopping time and the Borel-Cantelli lemma, we prove the existence and uniqueness of solutions. Then a global random attractor and the existence of a stationary measure are obtained via the Birkhoff ergodic theorem and the Chebyshev inequality.

Some limit theorems of delayed sums for rowwise conditionally independent stochastic arrays

ABSTRACTThis paper is concerned with the asymptotic property of delayed sums for rowwise conditionally independent stochastic arrays. The main technique of the proofing is to construct non negative random variables with one parameter and to apply the Borel–Cantelli lemma to obtaining almost everywhere convergence. The relevant results for non homogeneous Markov chains indexed by a tree are extended.

A kind of asymptotic properties of moving averages for Markov chains in Markovian environments

ABSTRACTConsider a Markov chain with finite alphabets. In this paper, we study the asymptotic properties of moving average, harmonic mean, and strong deviation theorems (limit theorems expressed by inequalities) of moving geometric average of random transition probabilities and the generalized entropy ergodic theorem for Markov chains in single infinite Markovian environments. It is shown that, under some mild conditions, the sequence of the generalized relative entropy density converges almost surely and in . The trick of the proofs is the construction of random variables with a parameter and the application of Borel–Cantelli lemma.

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

New sharp upper and lower bounds for conditional (given a σ-algebra A) probabilities of unions of events and for a generalization of the conditional Borel–Cantelli lemma are obtained. Averaging the left- and right-hand sides of the corresponding inequalities yields bounds better than those obtained by directly estimating the probabilities of events. An example is given. New generalizations of the conditional Borel–Cantelli lemma are also obtained. Averaging yields new versions of this lemma under conditions different from the classical ones.

Uncertain Zero-One Law and Convergence of Uncertain Sequence

This paper is concerned with situations in which the uncertain measure of an event can only be zero or one, and the uncertain zero-one laws are derived within the framework of uncertainty theory that can be seen as the counterpart of Kolmogorov zero-one law and Borel-Cantelli lemma, which can be used as a tool for solving some problems concerning almost sure convergence of uncertain sequence.

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

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

THE DUFFIN–SCHAEFFER‐TYPE CONJECTURES IN VARIOUS LOCAL FIELDS

This paper discovers a new phenomenon about the Duffin-Schaeffer conjecture, which claims that $\lambda(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$ if and only if $\sum_n\lambda({\mathcal E}_n)=\infty$, where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}/\mathbb{Z}$, \[ {\mathcal E}_n={\mathcal E}_n(\psi)=\bigcup_{m=1 \atop (m,n)=1}^n\big(\frac{m-\psi(n)}{n},\frac{m+\psi(n)}{n}\big), \] $\psi$ is any non-negative arithmetical function. Instead of studying $\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n$ we introduce an even fundamental object $\cup_{n=1}^{\infty}{\mathcal E}_n$ and conjecture there exists a universal constant $C>0$ such that \[\lambda(\bigcup_{n=1}^{\infty}{\mathcal E}_n)\geq C\min\{\sum_{n=1}^{\infty}\lambda({\mathcal E}_n),1\}.\] It is shown that this conjecture is equivalent to the Duffin-Schaeffer conjecture. Similar phenomena are found in the fields of $p$-adic numbers and formal Laurent series. As a byproduct, we answer conditionally a question of Haynes by showing that one can always use the quasi-independence on average method to deduce $\lambda(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$ as long as the Duffin-Schaeffer conjecture is true. We also show among several others that two conjectures of Haynes, Pollington and Velani are equivalent to the Duffin-Schaeffer conjecture, and introduce for the first time a weighted version of the second Borel-Cantelli lemma to the study of the Duffin-Schaeffer conjecture.

Maximum likelihood estimation for the drift parameter in diffusion processes

In this paper, parameter estimation for a stationary ergodic diffusion process with drift coefficient and diffusion coefficient is investigated. The likelihood function is given based on the Girsanov theorem and the existence of the maximum likelihood estimator is proved by applying the uniform ergodic theorem, Borel-Cantelli lemma and Rolle’s theorem. The consistency in probability of the estimator and asymptotic normality of the error of estimation are proved with the help of the uniform ergodic theorem, maximal inequality for martingale and Borel-Cantelli lemma.

Borel–Cantelli lemmas and extreme value theory for geometric Lorenz models

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 the convergence of rare events point processes to the standard Poisson process, which implies extreme value laws for observations maximised at generic points for geometric Lorenz maps. Further, we extend our extreme value laws to the associated flows.

On the strong convergence of weighted sums of widely dependent random variables

ABSTRACTFor widely dependent random variables, we present some results on the strong convergence of weighted sums, including results on almost surely (a.s.) and complete convergence. To this end, we verified some Borel–Cantelli lemmas of the widely dependent random variables. The above-mentioned random variables contain common negatively dependent random variables, some positively dependent random variables, and some others; therefore, the obtained results extend and improve some existing results.

COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF NEGATIVELY SUPERADDITIVE DEPENDENT RANDOM VARIABLES

In this paper we obtain complete convergences for weighted sums of negatively superadditive dependent random variables. These results generalize the corresponding ones for negatively associated random variables. P 1 i=1 P(jXi i Cj > †) 0. By Borel-Cantelli lemma, this implies that Xn ! C almost surely (a.s.). The converse is true if fXn;n ‚ 1g are independent. Hsu and Robbins (1947) also proved that the sequence of arithmetic means of i.i.d. random variables converges completely to the expected value if the variance of the summands is flnite. The complete convergence plays an important role in probability limit theory and mathematical statistics, especially, in establishing the con- vergence rate. A new kind of dependence structure called negatively superadditive dependence was introduced by Hu (2000) based on the class of superad- ditive functions. Superadditive structure functions have important relia- bility interpretations, which describe whether a system is more series-like

On estimation of probabilities of unions of events with applications to the Borel–Cantelli lemma

New upper and lower bounds for probabilities of unions of events are obtained. The inequalities proved are sharp and may turn into equalities. They make it possible to obtain new versions of both parts of the Borel–Cantelli lemma. The results remain valid for general (not necessarily probability) measure spaces.

Exponential stability of the exact and numerical solutions for neutral stochastic delay differential equations

This paper is concerned with pth moment and almost sure exponential stability of the exact and numerical solutions of neutral stochastic delay differential equations (NSDDEs). Moment exponential stability criteria of the continuous and discrete solutions are established by virtue of the Lyapunov method. Then the almost sure exponential stability criterion is derived by the Chebyshev inequality and the Borel–Cantelli lemma. We also examine conditions under which the numerical solution can reproduce the exponential stability of exact solution. It is shown that the linear growth condition is necessary for Euler–Maruyama (EM) method to maintain the moment exponential stability of the exact solution. If the drift coefficient of NSDDE satisfies the one-sided Lipschitz condition, EM method may break down, but we show that the backward EM (BEM) method can share the mean square exponential stability of the exact solution.

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 may be 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 parts of the Borel-Cantelli lemma.

From the First Borel–Cantelli Lemma to Poincaré's Recurrence Theorem

In this note, we prove the classical Poincaré's recurrence theorem by using the first Borel–Cantelli lemma.