On the strong law of large numbers for sequences of dependent random variables

New sufficient conditions for the applicability of the strong law of large numbers are established for sequences of random variables without the independence conditions. Results on strong stability of sums of dependent random variables are also obtained. No particular type of dependence between random variables of a sequence is assumed. Only conditions related to moments of random variables and their sums are used. It is shown that the results obtained are unimprovable in certain sense. These results are generalizations of some results of N. Etemadi proved under more restrictive conditions.     

On the strong law of large numbers for dependent random variables

Let {X _n} _n ^=1/∞ be a sequence of random variables with partial sumsS _n, and let {ie241-1} be the σ-algebra generated byX ₁,…,X _n. Letf be a function fromR toR and suppose {ie241-2}. Under conditions off and moment conditions on theX' _ns, we show thatS _n/n converges a.e. (almost everywhere). We give several applications of this result. Research supported by N.S.F. Grant MCS 77-26809     

On the applicability conditions of the strong law of large numbers for sequences of independent random variables

We investigate relationship between Kolmogorov–s condition and Petrov–s condition in theorems on the strong law of large numbers for a sequence of independent random variables X ₁, X ₂, … with finite variances. The convergence (S _n − ES _n )/n → 0 holds a.s. (here, S _n = Σ _k=1 ⁿ X _k ), provided that Σ _n=1^∞ DX _n /n ² < ∞ (Kolmogorov’s condition) or DS _n = O(n ²/ψ(n)) for some positive non-decreasing function ψ(n) such that Σ1/(nψ(n)) < ∞ (Petrov’s condition). Kolmogorov’s condition is shown to follow from Petrov’s condition. Besides, under some additional restrictions, Petrov’s condition, in turn, follows from Kolmogorov’s condition.     

A strong law of large numbers for orthogonal random variables

A generalization of one theorem of K. Tandori is proved. A sufficient condition is derived for application of a strong law of large numbers to a sequence of orthogonal random variables, expressed in terms of the growth of sums of second moments of these variables.     

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.     

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.     

On the strong law of large numbers for a sequence of nonnegative random variables

New sufficient conditions are found for the applicability of the strong law of large numbers to a sequence of dependent nonnegative random variables with finite variances. Bibliography: 4 titles.     

On complete convergence and the strong law of large numbers for pairwise independent random variables

The complete convergence for the maximum partial sums of pairwise independent random variables is obtained. The Kolmogorov strong law of large numbers for pairwise i.i.d. random variables is also obtained. Similar results are established for the moving average processes of pairwise i.i.d. random variables and for pairwise independent random elements taking values in a Banach space.     

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．     

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.     

Blocks of orthogonal random variables and the strong law of large numbers

(X _k ),k=1,2,... — _k ² >1; (X _k ) , E(X _k X _t )=0 p k<>(p+1) (p,k,l=1, 2, ...) , , ,

A note on the strong law of large numbers for positively dependent random variables

We obtain strong laws of large numbers for sequences of random variables which are either pairwise positive quadrant dependent or associated. Our results imply extensions of Kolmogoron's classical strong law to the positively dependent case.     

A strong law of large numbers for arrays of rowwise negatively dependent random variables

A strong law of large numbers for arrays of rowwise negatively dependent random variables is obtained which relaxes the usual assumption of rowwise independence. The moment conditions of the main result are similar to previous results, and the stochastic bounded condition also provides a relaxation of the usual distributional assumptions.     

The strong law of large numbers for random processes

Analogs of the Kolmogorov, Zygmund-Martsenkevich, and Brunk-Prokhorov strong law of large numbers are proved for martingales with continuous parameter. A new generalization of the Brunk—Prokhorov strong law of large numbers is given for martingales with discrete times. Along with convergence almost everywhere, we also prove the average convergence.     

On the strong law of large numbers for delayed sums and random fields

A paper by Chow [3] contains (i.a.) a strong law for delayed sums, such that the length of the edge of the nth window equals n ^α for 0 < α < 1. In this paper we consider the kind of intermediate case when edges grow like n=L(n), where L is slowly varying at infinity, thus at a higher rate than any power less than one, but not quite at a linear rate. The typical example one should have in mind is L(n) = log n. The main focus of the present paper is on random field versions of such strong laws.     

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].