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.     

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.     

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.     

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.     

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 law of large numbers for almost periodically correlated processes

This note contains two simple observations concerning the weak law of large numbers for almost periodically correlated processes.

     

LARGE DEVIATIONS AND MODERATE DEVIATIONS FOR m-NEGATIVELY ASSOCIATED RANDOM VARIABLES

M-negatively associated random variables,which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its par- ticular case,are introduced and studied.Large deviation principles and moderate devi- ation upper bounds for stationary m-negatively associated random variables are proved. Kolmogorov-type and Marcinkiewicz-type strong laws of large numbers as well as the three series theorem for m-negatively associated random variables are also given.     

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.     

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.     

Law of large numbers for increasing subsequences of random permutations

Let the random variable Z_n,k denote the number of increasing subsequences of length k in a random permutation from S_n, the symmetric group of permutations of {1,…,n}. We show that Var(Z) = o((EZ)²) as n → ∞ if and only if . In particular then, the weak law of large numbers holds for Z if ; that is, We also show the following approximation result for the uniform measure U_n on S_n. Define the probability measure μ on S_n by where U denotes the uniform measure on the subset of permutations that contain the increasing subsequence {x₁,x₂,…,x}. Then the weak law of large numbers holds for Z if and only if where ∣∣˙∣∣ denotes the total variation norm. In particular then, (*) holds if . In order to evaluate the asymptotic behavior of the second moment, we need to analyze occupation times of certain conditioned two‐dimensional random walks. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

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．     

A mixed random walk on nonnegative matrices: A law of large numbers

In this paper a mixed random walk on nonnegative matrices has been studied. Under reasonable conditions, existence of a unique invariant probability measure and a law of large numbers have been established for such walks.     

On the strong limit theorems for double arrays of blockwise M-dependent random variables

For a double array of blockwise M-dependent random variables {X _mn ,m ?? 1, n ?? 1}, strong laws of large numbers are established for double sums ?? _i=1 ^m ?? _j=1 ⁿ X _ij , m ?? 1, n ?? 1. The main results are obtained for (i) random variables {X _mn ,m ?? 1, n ?? 1} being non-identically distributed but satisfy a condition on the summability condition for the moments and (ii) random variables {X _mn ,m ?? 1, n ?? 1} being stochastically dominated. The result in Case (i) generalizes the main result of Móricz et al. [J. Theoret. Probab., 21, 660?C671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [Ann. Probab., 6, 469?C482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.     

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.     

END随机变量阵列加权和的完全收敛性

利用END变量的R0senthal型矩不等式,研究了END随机阵列加权和的完全收敛性,给出了证明完全收敛性的一些充分条件.另外,还给出了证明完全收敛性的一个必要条件.所得结果推广了独立变量和若干相依变量的相应结果.     

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].     

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.