Rates of convergence for the laws of large numbers for independent Banach-valued random variables

In this paper, results of Lai, Heyde, and Rohatgi concerning the convergence rates for the laws of large numbers are extended for the case of independent random variables taking values in a separable Banach space.     

On the laws of large numbers for nonnegative random variables

Strong laws of large numbers concerning nonnegative random variables are obtained and then they are utilized to establish stability results, among other things, for sums of pairwise independent random variables and the range of random walks.     

Weak laws of large numbers for arrays of dependent random variables

In this paper, we establish some weak laws of large numbers for arrays of dependent random variables satisfying the conditions of a kind of uniform integrability. Our results extend and improve the corresponding ones.     

Exponential inequalities for associated random variables and strong laws of large numbers

Some exponential inequalities for partial sums of associated random variables are established. These inequalities improve the corresponding results obtained by Ioannides and Roussas (1999), and Oliveira (2005). As application, some strong laws of large numbers are given. For the case of geometrically decreasing covariances, we obtain the rate of convergence n-1/2(log log n)1/2(logn) which is close to the optimal achievable convergence rate for independent random variables under an iterated logarithm, while Ioannides and Roussas (1999), and Oliveira (2005) only got n-1/3(logn)2/3 and n-1/3(logn)5/3, separately.     

Mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables

Under some conditions of uniform integrability and appropriate conditions, mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables are obtained. Our results extend and improve the results of [H.S. Sung, S. Lisawadi, A. Volodin, Weak laws of large numbers for arrays under a condition of uniform integrability, J. Korean Math. Soc. 45 (2008) 289-300] and [M. Ordóñez Cabrera, A. Volodin, Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability, J. Math. Anal. Appl. 305 (2005) 644-658].     

次线性期望下Pareto型随机变量的大数律

本文对满足Pareto分布的随机变量建立了一些大数律,从而将经典概率空间中的相关结论推广到次线性期望空间中.基于Pareto分布,获得了一些独立随机变量序列加权和的弱大数律和强大数律.     

On the laws of large numbers for double arrays of independent random elements in Banach spaces

For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ≥ 1,n ≥ 1 are equivalent.Both the identically distributed and the nonidentically distributed cases are treated.In the main theorems,no assumptions are made concerning the geometry of the underlying Banach space.These theorems are applied to obtain Kolmogorov,Brunk–Chung,and Marcinkiewicz–Zygmund type strong laws of large numbers for double sums in Rademacher type p(1 ≤ p ≤ 2) Banach spaces.     

Kolmogorov's strong law of large numbers for fuzzy random variables

In this paper, Kolmogorov's strong law of large numbers for sums of independent and level-wise identically distributed fuzzy random variables is obtained.     

Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations

In this paper, we present some multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, which extend some previous results.     

A strong law of large numbers for fuzzy random variables

A limit of a sequence of fuzzy numbers is defined and its some properties are shown. Based on these concept and properties, an independent sequence of fuzzy random variables is considered and a strong law of large numbers for fuzzy random variables is shown.     

The strong law of large numbers for pairwise negatively dependent random variables

Necessary and sufficient conditions for the validity of the strong law of large numbers for pairwise negatively dependent random variables with infinite means are formulated.     

Inequalities and strong laws of large numbers for random allocations

Summary Moment inqualities and strong laws of large numbers are proved for random allocations of balls into boxes. Random broken lines and random step lines are constructed using partial sums of i.i.d. random variables that are modified by random allocations. Functional limit theorems for such random processes are obtained.     

关于M值随机序列的一个普遍成立的强大数定理

利用区间剖分法构造几乎处处收敛的鞅,得到了一个对任意M-值随机变量序列普遍成立的强极限定理,作为推论得到一个精细的Borel—Cantelli引理．     

Laws of large numbers of negatively correlated random variables for capacities

Our aim is to present some limit theorems for capacities.We consider a sequence of pairwise negatively correlated random variables.We obtain laws of large numbers for upper probabilities and 2-alternating capacities,using some results in the classical probability theory and a non-additive version of Chebyshev’s inequality and Boral-Contelli lemma for capacities.     

Concentration inequalities and laws of large numbers under epistemic and regular irrelevance

his paper presents concentration inequalities and laws of large numbers under weak assumptions of irrelevance that are expressed using lower and upper expectations. The results build upon De Cooman and Miranda’s recent inequalities and laws of large numbers. The proofs indicate connections between the theory of martingales and concepts of epistemic and regular irrelevance.     

Strong law of large numbers for pair-wise extended lower/upper negatively dependent random variables

We establish some strong limit theorems for a sequence of pair-wise extended lower/upper negatively dependent random variables and give some new examples of dependent random variables.     

Uniform laws of large numbers for set-valued mappings and subdifferentials of random functions

We derive a uniform (strong) Law of Large Numbers (LLN) for random set-valued mappings. The result can be viewed as an extension of both, a uniform LLN for random functions and LLN for random sets. We apply the established results to a consistency analysis of stationary points of sample average approximations of nonsmooth stochastic programs.     

Some remarks on the strong law of large numbers for Banach-space valued weakly integrable random variables

The purpose of this paper is to show the equivalence of almost sure convergence of S_n/n, n ≥ 1 and lim sup_n→∞S_n/n < ∞ a.e., where S_n = X₁ + X₂ + … + X_n and X₁, X₂,… are independent identically distributed random elements in a separable Banach space with EX₁ < ∞. This result disproves a result of Pop-Stojanovic [8].     

One-sided limit theorems for sums of multidimensional indexed random variables

Let {X,X _n,nZ ₊ ^d } be a sequence of independent and identically distributed random variables and {a _n ,n Z ₊ ^d } be a sequence of constants. We examine the almost sure limiting behavior of weighted partial sums of the form _|n|N a _n X _n . Suppose further that eitherEX=0 orE|X|=. In most situations these normalized partial sums fail to have a limit, no matter which normalizing sequence we choose. Thus, the investigation lends itself to the study of the limit inferior and limit superior of these sequences. On the way to proving results of this type we first establish several weak laws. These weak laws prove to be of great value in establishing generalized laws of the iterated logarithm.