Borel-Cantelli Lemma Research Articles

A recurrence-type strong Borel–Cantelli lemma for Axiom A diffeomorphisms

Abstract Let $(X,\mu ,T,d)$ be a metric measure-preserving dynamical system such that three-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence $(M_k)$ that converges to $0$ slowly enough, we obtain a strong dynamical Borel–Cantelli result for recurrence, that is, for $\mu $ -almost every $x\in X$ , $$ \begin{align*} \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} \mu(B_k(x))} = 1, \end{align*} $$ where $\mu (B_k(x)) = M_k$ . In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.

On the extension of quadrant dependence

In this short note, it is propounded an extension for quadrant dependence, and shown that some of the original proprieties of this popular concept remain valid, while others are necessarily generalized. A second Borel–Cantelli lemma due to Petrov (Statist. Probab. Lett. 58: 283–286, 2002) is revisited for events enjoying this new dependence notion and demonstrated by means of simpler arguments.

Almost Sure Central Limit Theorem for Error Variance Estimator in Pth-Order Nonlinear Autoregressive Processes

In this paper, under some suitable assumptions, using the Taylor expansion, Borel–Cantelli lemma and the almost sure central limit theorem for independent random variables, the almost sure central limit theorem for error variance estimator in the pth-order nonlinear autoregressive processes with independent and identical distributed errors was established. Four examples, first-order autoregressive processes, self-exciting threshold autoregressive processes, threshold-exponential AR progresses and multilayer perceptrons progress, are given to verify the results.

Boundary dynamics for holomorphic sequences, non-autonomous dynamical systems and wandering domains

There are many classical results, related to the Denjoy–Wolff theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, we address such questions in the very general setting of sequences (Fn) of holomorphic maps between simply connected domains. We show that, while some classical results can be generalised, with an interesting dependence on the geometry of the domains, a much richer variety of behaviours is possible. One of our main results is new even in the classical setting.Our methods apply in particular to non-autonomous dynamical systems, when (Fn) are forward compositions of holomorphic maps, and to the study of wandering domains in holomorphic dynamics.The proofs use techniques from geometric function theory, measure theory and ergodic theory, and the construction of examples involves a ‘weak independence’ version of the second Borel–Cantelli lemma and the concept from ergodic theory of ‘shrinking targets’.

Almost sure exponential synchronization of multilayer complex networks with Markovian switching via aperiodically intermittent discrete observa- tion noise

<p>This paper is concerned with almost sure exponential synchronization of multilayer complex networks with Markovian switching via aperiodically intermittent discrete observation noise. First, Markovian switching and multilayer interaction factors are taken into account simultaneously, which make our system more general compared with the existing literature. Meanwhile, the network architecture may be undirected or directed, and consequently, the adjacency matrix is symmetrical and asymmetrical. Second, the control strategy is based on aperiodically intermittent discrete observation noise, where the average control rate is integrated to depict the distributions of work/rest intervals of the control strategy from an overall perspective. Third, different from the work about $ p $th moment exponential synchronization of network systems, by utilizing M-matrix theory and various stochastic analysis techniques including the Itô formula, the Gronwall inequality, and the Borel-Cantelli lemma, some criteria on almost sure exponential synchronization of multilayer complex networks with Markovian switching have been constructed and the upper bound of the duration time has been also estimated. Finally, several numerical simulations are exhibited to validate the effectiveness and feasibility of our analytical findings.</p>

Stochastic McKean–Vlasov equations with Lévy noise: Existence, attractiveness and stability

This work considers McKean–Vlasov equations driven by Lévy noise in which the coefficients rely not only on the state of the unknown process but also on its probability distribution, and researches the well-posed, attractiveness and stability of solutions with resolvent operator theory. Under global Lipschitz condition, the well-posed of mild solutions is first studied for McKean–Vlasov integro-differential equations by using contraction mapping principle. Then according to Cauchy–Schwartz inequality and Itô isometry formula, one gains global attracting set and quasi-invariant set of solutions of McKean–Vlasov integro-differential equations and also obtained sufficient conditions of stability with continuous dependence on the coefficient of solutions. Moreover, one attains mean-square stability of solutions by Gronwall inequality and almost sure stability by applying Borel–Cantelli lemma and Chebyshev inequality for the aforementioned equation. Finally two examples are presented to verify our results.

The asymptotic equipartition property for a special Markov random field

The asymptotic equipartition property (AEP) plays an important role in establishing lossless source coding theorems and asymptotic coding theorems through the concepts of typical sets and typical sequences in information theory. In this paper, we study the generalized asymptotic equipartition property in the form of moving average for N bifurcating Markov chains indexed by an N-branch Cayley tree, which is a special case of Markov Urandom fields. Firstly, we construct a class of random variables containing a parameter with means of 1, and establish a strong limit theorem for the moving average of multivariate functions of such chains using the Borel–Cantelli lemma. Secondly, we present the strong law of large numbers for the frequencies of occurrence of states of the moving average, as well as the generalized asymptotic equipartition property for N bifurcating Markov chains indexed by an N-branch Cayley tree. As corollaries, we also generalize some known results.

On asymptotic properties of spacings

In this work, we investigate spacings based on order statistics obtained from continuous distribution functions. At the beginning of the paper, we present distributional results for spacings and a method of classification of distributions according to their tails. Then we use this method to derive asymptotic results for spacings. By applying some special versions of Borel–Cantelli lemma, we obtain their strong limit results. At the end of the paper, we present some illustrative examples.

Dynamic analysis and optimal control of a stochastic information spreading model considering super-spreader and implicit exposer with random parametric perturbations

Environmental factors in social systems affect information spreading at all times. This paper proposes a stochastic S2EIR model that considers the presence of super-spreaders and implicit exposers in information spreading, as well as the stochastic perturbation of model parameters. The existence of a global positive solution using the Itô′s formula is then demonstrated. Sufficient conditions for information disappearance and smooth distribution of information are calculated by using the Borel–Cantelli lemma and the strong law of large numbers. Furthermore, the optimal control strategy for the stochastic model is proposed using the Hamiltonian function. The results of the theoretical analysis are supported by numerical simulations and compared to the parameter variations of the deterministic model. The results of this study indicate that white noise facilitates the spread of information. The intensity of perturbation is proportional to the fluctuation of information spreading. Controlling random parameters can effectively facilitate the spread of information. For positive information, the randomness and complexity of the social system should be utilized to increase the spread of information. In contrast, for negative information, randomness in the social system should be suppressed to the greatest extent possible to limit information dissemination.

Impulsive-Based Almost Surely Synchronization for Neural Network Systems Subject to Deception Attacks.

This article is dedicated to investigating the impulsive-based almost surely synchronization issue of neural network systems (NSSs) with quality-of-service constraints. First, the communication network considered suffers from random double deception attacks, which are modeled as a nonlinear function and a desynchronizing impulse sequence, respectively. Meanwhile, the impulsive instants and impulsive gains are randomly and only their expectations are available. Second, by taking two different types of random deception attacks into consideration, a novel mathematical model for vulnerable NSSs is constructed. Then, almost surely synchronization criteria are established by using Borel-Cantelli lemma. Furthermore, based on the derived strong and weak sufficient conditions, the almost surely synchronization of NSSs is achieved. Finally, the section of numerical example is shown to illustrate the effectiveness of the proposed method.

Dynamical Borel–Cantelli lemma for recurrence under Lipschitz twists

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 laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the zero–one laws for the two problems.

Razumikhin Theorems on Polynomial Stability of Neutral Stochastic Pantograph Differential Equations with Markovian Switching

This paper investigates the polynomial stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEsMS). Firstly, under the local Lipschitz condition and a more general nonlinear growth condition, the existence and uniqueness of the global solution to the addressed NSPDEsMS is considered. Secondly, by adopting the Razumikhin approach, one new criterion on the qth moment polynomial stability of NSPDEsMS is established. Moreover, combining with the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure polynomial stability of NSPDEsMS is examined. The results derived in this paper generalize the previous relevant ones. Finally, two examples are provided to illustrate the effectiveness of the theoretical work.

A Note on Exponential Stability for Numerical Solution of Neutral Stochastic Functional Differential Equations

This paper examines the numerical solutions of the neutral stochastic functional differential equation. This study establishes the discrete stochastic Razumikhin-type theorem to investigate the exponential stability in the mean square sense of the Euler–Maruyama numerical solution to this equation. In addition, the Borel–Cantelli lemma and the stochastic analysis theory are incorporated to discuss the almost sure exponential stability for this numerical solution of such equations.

Boundedness analysis of stochastic delay differential equations with Lévy noise

The present paper is concerned with the mean square asymptotic boundedness and twice power almost surely asymptotic boundedness of stochastic delay differential equations driven by Lévy noise. First, mean square asymptotic boundedness criteria of the solutions are established by the method of reduction and the generalized Itô formula. Then, based on the Chebyshev inequality and the Borel–Cantelli lemma, the twice power almost surely asymptotic boundedness criteria are also derived for the addressed equations. Finally, a example is provided to demonstrate the validity of the proposed results.

Nonparametric estimation of periodic signal disturbed by α-stable noises

The paper is concerned with the nonparametric estimation problem for periodic signal disturbed by α-stable noise based on a periodic kernel. The consistency and the inconsistency of the nonparametric estimator separately in terms of the range of α, i.e. α ∈ (1,2) and α ∈ (0,1] are discussed by using Markov inequality and the moment inequalities for stable stochastic integrals. Then, we state the strong consistency of the nonparametric estimator by using the triangular kernel and the Borel-Cantelli lemma when α ∈ (1,2). Besides, the asymptotic distributions of the nonparametric estimator are studied by applying the inner clock property for the α-stable stochastic integral. Taylor's formula and Slutsky's theorem. Finally, a numerical example is introduced to illustrate our theory

On strong forms of the Borel-Cantelli lemma and dynamical systems with polynomial decays of correlations

Strong forms of the Borel - Cantelli lemma are variants of the strong law of large numbers for sums of the indicators of events such that the series from probabilities of these events diverges. These sums are centered at means and normalized by some function from means. In this paper, we derive new strong forms of the Borel - Cantelli lemma under wider restrictions on variations of increments of sums than it was done earlier. Strong forms are commonly used for investigations of properties of dynamical systems. We apply our results to describe properties of some measure preserving expanding maps of [0, 1] with a fixed point at zero. Such results can be proved for similar multidimensional maps as well. Keywords: the Borel - Cantelli lemma, the quantitative Borel - Cantelli lemma, strong forms of the Borel - Cantelli lemma, sums of indicators of events, strong law of large num bers, almost surely convergence, dynamical systems, polynomial decay of correlations.

On a strong form of the Borel-Cantelli lemma

The strong form of the Borel-Cantelli lemma is a variant of the strong law of large numbers for sums of the indicators of events. These sums are centered at the mean and normalized by some function from sums of probabilities of events. The series from probabilities is assumed to be divergent. In this paper, we derive new strong forms of the Borel-Cantelli lemma with smaller normalizing sequences than it was before. Conditions on variations of increments of indicators become stronger. We give examples in which these conditions hold.

On the Borel-Cantelli Lemmas, the Erdős-Rényi Theorem, and the Kochen-Stone Theorem

In this paper we present a quantitative analysis of the first and second Borel-Cantelli Lemmas and of two of their generalisations: the Erdős–Rényi Theorem, and the Kochen-Stone Theorem. We will see that the first three results have direct quantitative formulations, giving an explicit relationship between quantitative formulations of the assumptions and the conclusion. For the Kochen-Stone theorem, however, we can show that the numerical bounds of a direct quantitative formulation are not computable in general. Nonetheless, we obtain a quantitative formulation of the Kochen-Stone Theorem using Tao's notion of metastability.

Exponential stability of θ-EM method for nonlinear stochastic Volterra integro-differential equations

Mean square exponential stability of both exact solutions and the corresponding θ-EM method for stochastic Volterra integro-differential equations are investigated in this paper. For 12<θ≤1, we prove that both exact solutions and the corresponding θ-EM method for stochastic Volterra integro-differential equations are mean square exponentially stable under the Khasminskii-type conditions. If 0≤θ≤12, θ-EM method is mean square exponentially stable under the Khasminskii-type condition plus linear growth condition on f. By using Chebyshev inequality and Borel-Cantelli lemma, we can also prove that θ-EM method is almost surely exponentially stable. An example is provided to support our conclusions.

Stability analysis of theθ-method for hybrid neutral stochastic functional differential equations with jumps

• Few results on the stability of numerical methods for the neutral stochastic functional differential equations with Markovian switching and jumps, results obtained in this manuscript can enrich this area. • Illustrate the almost sure exponential stability and the mean-square exponential stability of the trivial solution. • Show the θ -method can preserve the almost sure exponential stability and the mean-square exponential stability of the trivial solution under some appropriate condition. • The almost sure exponential stability of the θ -method is obtained by the discrete semi-martingale convergence theorem without resorting to using the Borel-Cantelli lemma and the Chebyshev inequality. Few results seems to be known about the stability of numerical methods for the hybrid neutral stochastic functional differential equations with jumps (also known as the neutral stochastic functional differential equations with Markovian switching and jumps (NSFDEwMJs)). This paper mainly investigates the exponential stability (both the almost sure and the mean-square exponential stability) of the θ -method for NSFDEwMJs. Precisely, it is first illustrated that the trivial solution of the NSFDEwMJ is almost surely and mean-square exponentially stable. It is then shown that the θ -method can preserve the same conclusions of the trivial solution. Numerical examples are demonstrated to illustrate the obtained results.