An exponential inequality for widely orthant dependent random variables and its application to a first-order autoregressive model

Probability Inequalities for Sums of Independent Random Variables

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

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.

This paper mainly studies the strong convergence properties for weighted sums of extended negatively dependent (END, for short) random variables. Some sufficient conditions to prove the strong law of large numbers for weighted sums of END random variables are provided. In particular, the authors obtain the weighted version of Kolmogorov type strong law of large numbers for END random variables as a product. The results that the authors obtained generalize the corresponding ones for independent random variables and some dependent random variables. As an application, the authors investigate the errors-in-variables (EV, for short) regression models and establish the strong consistency for the least square estimators. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analysed for illustration.

?Strong laws of large numbers which are useful in the theory and applications of stochastic processes are obtained for sequences of independent random elements in separable normed linear spaces. The hypotheses for these results lie between those for the identically distributed case and the independent non-identieally distributed case. These results and other strong and weak laws of large numbers for separable normed linear spaces can be extended to separable Freshet spa?es. Finally, the results are applied to separable Wiener processes on [0, 1] and on [0, oo).

Let {<TEX>$X_n,n{\geq}1$</TEX>} be a sequence of negatively superadditive dependent random variables. In the paper, we study the strong law of large numbers for general weighted sums <TEX>${\frac{1}{g(n)}}{\sum_{i=1}^{n}}{\frac{X_i}{h(i)}}$</TEX> of negatively superadditive dependent random variables with non-identical distribution. Some sufficient conditions for the strong law of large numbers are provided. As applications, the Kolmogorov strong law of large numbers and Marcinkiewicz-Zygmund strong law of large numbers for negatively superadditive dependent random variables are obtained. Our results generalize the corresponding ones for independent random variables and negatively associated random variables.

In the paper, the Marcinkiewicz–Zygmund type moment inequality for extended negatively dependent (END, in short) random variables is established. Under some suitable conditions of uniform integrability, the $$L_r$$ convergence, weak law of large numbers and strong law of large numbers for usual normed sums and weighted sums of arrays of rowwise END random variables are investigated by using the Marcinkiewicz–Zygmund type moment inequality. In addition, some applications of the $$L_r$$ convergence, weak and strong laws of large numbers to nonparametric regression models based on END errors are provided. The results obtained in the paper generalize or improve some corresponding ones for negatively associated random variables and negatively orthant dependent random variables.

Some exponential probability inequalities for widely negative orthant dependent (WNOD, in short) random variables are established, which can be treated as very important roles to prove the strong law of large numbers among others in probability theory and mathematical statistics. By using the exponential probability inequalities, we study the complete convergence for arrays of rowwise WNOD random variables. As an application, the Marcinkiewicz–Zygmund type strong law of large numbers is obtained. In addition, the complete moment convergence for arrays of rowwise WNOD random variables is studied by using the exponential probability inequality and complete convergence that we established.

In this paper, we study strong laws of large numbers for weighted sums of extended negatively dependent random variables under sub-linear expectation space. As an application, several results on strong laws of large numbers of with the condition of and for the double arrays of positive real numbers and sequences of extended negatively dependent random variables have been established in sub-linear expectations. The main results obtained in this article are the extensions of strong laws of large numbers for weighted sums of negatively dependent random variables under the traditional probability space.

In this work, we obtain two main theorems of strong law of large numbers for a 2-dimensional array of random variables. The first theorem is the strong law of large numbers for pairwise negatively dependent random variables which are not necessary identically distributed. The second theorem, we obtain strong law of large numbers for the difference of random variables which independent and identically distributed conditions are regarded. In this study, we use the limit as m×n tends to infinity instead of using the limit as m,n tends to infinity when m,n are natural numbers which is stronger.

This paper establishes complete convergence for weighted sums and the Marcinkiewicz–Zygmund-type strong law of large numbers for sequences of negatively associated and identically distributed random variables $$\{X,X_n,n\ge 1\}$$ with general normalizing constants under a moment condition that $$ER(X)<\infty $$ , where $$R(\cdot )$$ is a regularly varying function. The result is new even when the random variables are independent and identically distributed (i.i.d.), and a special case of this result comes close to a solution to an open question raised by Chen and Sung (Stat Probab Lett 92:45–52, 2014). The proof exploits some properties of slowly varying functions and the de Bruijn conjugates. A counterpart of the main result obtained by Martikainen (J Math Sci 75(5):1944–1946, 1995) on the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise i.i.d. random variables is also presented. Two illustrative examples are provided, including a strong law of large numbers for pairwise negatively dependent random variables which have the same distribution as the random variable appearing in the St. Petersburg game.

We study strong laws of large numbers in a non-linear framework based on conditional sub-additive expectations and conditional sub-additive capacities. Using an axiomatic approach to conditional sub-additive expectation, we establish a conditional Hájek–Rényi-type maximal inequality assuming a general conditional Kolmogorov-type maximal inequality but without imposing any weak dependence structure on the underlying sequence. As a consequence, we derive a general conditional strong law of large numbers. Finally, we introduce a notion of conditional negative dependence under sub-additive expectations and obtain the corresponding conditional Kolmogorov-type maximal inequality, leading to a conditional strong law of large numbers for conditionally negatively dependent random variables.

In this paper, the complete convergence and the complete moment convergence for extended negatively dependent (END, in short) random variables without identical distribution are investigated. Under some suitable conditions, the equivalence between the moment of random variables and the complete convergence is established. In addition, the equivalence between the moment of random variables and the complete moment convergence is also proved. As applications, the Marcinkiewicz-Zygmund-type strong law of large numbers and the Baum-Katz-type result for END random variables are established. The results obtained in this paper extend the corresponding ones for independent random variables and some dependent random variables.

Let {Xn, n ⩾ 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {bni, 1 ⩽ i ⩽ n, n ⩾ 1} be an array of real numbers. In this article, we study the strong law of large numbers for the weighted sums ∑ni = 1bniXi without identical distribution. We present some sufficient conditions to prove the strong law of large numbers. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for NSD random variables is obtained. In addition, the complete convergence for the weighted sums of NSD random variables is established. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.

In this work, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of widely orthant dependent random variables is established under mild conditions including infinite variance. As applications, some new results such as the Cesàro strong law of large numbers, the strong consistency of the least squares estimator in multiple linear regression models, and the strong consistency of the wavelet estimator in nonparametric regression models are presented. Simulation studies are also provided to support the theoretical results.