Quantitative strong laws of large numbers for random variables with double indices

We find necessary and sufficient conditions for the weighted strong law of large numbers for independent random variables with multidimensional indices belonging to some sector.

In this contribution, we introduce gradual numbers and intervals of gradual numbers to be able to prove the strong law of large numbers for random variables that are valued by such elements. The results show a novel point of view in the investigation of strong law of large numbers for random variables that are valued by non-exactly specified numbers.

The objective of this paper is to derive some limit theorems of fuzzy random variables under the extension principle associated with continuous Archimedean triangular norms (t-norms). First of all, some convergence theorems for the sum of fuzzy random variables in chance measure and expected value are proved respectively based on the arithmetics of continuous Archimedean triangular norms. Then, a law of large numbers for fuzzy random variables is established by using the obtained convergence theorems. The results of the derived law of large numbers can degenerate to the strong laws of large numbers for random variables and fuzzy variables, respectively.

In this article, the strong law of large numbers for weighted sums of asymptotically almost negatively associated (AANA, in short) random variables is obtained. Some sufficient conditions for the strong law of large numbers of random variables are presented. In addition, the results of the paper generalize and improve earlier ones of Chung (Am J Math 69:189–192, 1947) and Teicher (Proc Natl Acad Sci USA 59:705–707, 1968).

On the strong laws of large numbers for weighted sums of random variables

Let \({\{X_n, n \geq1 \}}\) be a sequence of random variables and {bn, n ≥ 1} a nondecreasing sequence of positive constants. No assumptions are imposed on the joint distributions of the random variables. Some sufficient conditions are given under which \({\lim_{n\to \infty}\sum_{i=1}^n X_i/b_n=0}\) almost surely. Necessary conditions for the strong law of large numbers are also given.

In this paper, the Chung’s strong law of large numbers is generalized to the random variables which do not need the condition of independence, while the sequence of Borel functions verifies some conditions weaker than that in Chung’s theorem. Some convergence theorems for martingale difference sequence such as Lp martingale difference sequence are the particular cases of results achieved in this paper. Finally, the convergence theorem for A-summability of sequence of random variables is proved, where A is a suitable real infinite matrix.

Strongly consistent estimates are shown, via relative frequency, for the probability of white balls inside a dichotomous urn when such a probability is an arbitrary unknown continuous time-dependent function over a bounded time interval. The asymptotic behaviour of relative frequency is studied in a nonstationary context using a Riemann-Dini type theorem for strong law of large numbers of random variables with arbitrarily different expectations; furthermore, the theoretical results concerning the strong law of large numbers can be applied for estimating the mean function of an unknown form of a general nonstationary process.

Law of large numbers for weak Fubini-independent random variables under capacities without sub-additivity

Based on credibility theory, this paper studies the convergent properties about the sum of fuzzy variables. We first propose several convergence theorems in credibility and in expectation on the sum of fuzzy variables, and then we establish a strong law of large numbers for fuzzy variables, finally, a theorem similar to the law of large numbers for random variables is also obtained.

Consider a set of independent identically distributed random variables indexed by $Z^d_+$, the positive integer $d$-dimensional lattice points, $d \geqq 2$. The classical Kolmogorov-Marcinkiewicz strong law of large numbers is generalized to this case. Also, convergence rates in the law of large numbers are derived, i.e., the rate of convergence to zero of, for example, the tail probabilities of the sample sums is determined.

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General introduction.- Mathematical preliminaries.- Random elements in separable metric spaces.- Laws of large numbers for random variables and separable Hilbert spaces.- Strong laws of large numbers for normed linear spaces.- Weak laws of large numbers for normed linear spaces.- Laws of large numbers for Frechet spaces.- Some applications.

General introduction.- Mathematical preliminaries.- Random elements in separable metric spaces.- Laws of large numbers for random variables and separable Hilbert spaces.- Strong laws of large numbers for normed linear spaces.- Weak laws of large numbers for normed linear spaces.- Laws of large numbers for Frechet spaces.- Some applications.

We extend to random fields case, the results of Woyczynski, who proved Brunk′s type strong law of large numbers (SLLNs) for 𝔹‐valued random vectors under geometric assumptions. Also, we give probabilistic requirements for above‐mentioned SLLN, related to results obtained by Acosta as well as necessary and sufficient probabilistic conditions for the geometry of Banach space associated to the strong and weak law of large numbers for multidimensionally indexed random vectors.