## Random Variables

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### Przemysław Matuła - One of the best experts on this subject based on the ideXlab platform.

• ##### A note on the almost sure convergence of sums of negatively dependent Random Variables
Statistics & Probability Letters, 1992
Co-Authors: Przemysław Matuła
Abstract:

Abstract We study the almost sure convergence of sums of negatively dependent Random Variables, in particular, the classical strong law of large numbers for independent distributed Random Variables is generalized to the case of pairwise negative quadrant dependent Random Variables. We also extend the three series theorem to the case of negatively associated Random Variables.

### Xiangchen Wang - One of the best experts on this subject based on the ideXlab platform.

• ##### complete convergence and almost sure convergence of weighted sums of Random Variables
Journal of Theoretical Probability, 1995
Co-Authors: Bhaskara M Rao, Tiefeng Jiang, Xiangchen Wang
Abstract:

Letr>1. For eachn≥1, let {X nk , −∞Random Variables. We provide some very relaxed conditions which will guarantee $$\Sigma _{n \geqslant 1} n^{r - 2} P\{ |\Sigma _{k = - \infty }^\infty X_{nk} | \geqslant \varepsilon \}< \infty$$ for every e>0. This result is used to establish some results on complete convergence for weighted sums of independent Random Variables. The main idea is that we devise an effetive way of combining a certain maximal inequality of Hoffmann-Jorgensen and rates of convergence in the Weak Law of Large Numbers to establish results on complete convergence of weighted sums of independent Random Variables. New results as well as simple new proofs of known ones illustrate the usefulness of our method in this context. We show further that this approach can be used in the study of almost sure convergence for weighted sums of independent Random Variables. Convergence rates in the almost sure convergence of some summability methods ofiid Random Variables are also established.

### Siddharth Muthukrishnan - One of the best experts on this subject based on the ideXlab platform.

• ##### entropy power inequality for a family of discrete Random Variables
International Symposium on Information Theory, 2011
Co-Authors: Naresh Sharma, Siddharth Muthukrishnan
Abstract:

It is known that the Entropy Power Inequality (EPI) always holds if the Random Variables have density. Not much work has been done to identify discrete distributions for which the inequality holds with the differential entropy replaced by the discrete entropy. Harremoes and Vignat showed that it holds for the pair (B(m, p),B(n, p)), m, n ∈ ℕ, (where B(n, p) is a Binomial distribution with n trials each with success probability p) for p = 0.5. In this paper, we considerably expand the set of Binomial distributions for which the inequality holds and, in particular, identify n 0 (p) such that for all m, n ≥ n 0 (p), the EPI holds for (B(m, p),B(n, p)). We further show that the EPI holds for the discrete Random Variables that can be expressed as the sum of n independent and identically distributed (IID) discrete Random Variables for large n.

• ##### Entropy power inequality for a family of discrete Random Variables
2011 IEEE International Symposium on Information Theory Proceedings, 2011
Co-Authors: Naresh Sharma, Smarajit Das, Siddharth Muthukrishnan
Abstract:

It is known that the Entropy Power Inequality (EPI) always holds if the Random Variables have density. Not much work has been done to identify discrete distributions for which the inequality holds with the differential entropy replaced by the discrete entropy. Harremo\"{e}s and Vignat showed that it holds for the pair (B(m,p), B(n,p)), m,n \in \mathbb{N}, (where B(n,p) is a Binomial distribution with n trials each with success probability p) for p = 0.5. In this paper, we considerably expand the set of Binomial distributions for which the inequality holds and, in particular, identify n_0(p) such that for all m,n \geq n_0(p), the EPI holds for (B(m,p), B(n,p)). We further show that the EPI holds for the discrete Random Variables that can be expressed as the sum of n independent identical distributed (IID) discrete Random Variables for large n.

### Varun Jog - One of the best experts on this subject based on the ideXlab platform.

• ##### an entropy inequality for symmetric Random Variables
International Symposium on Information Theory, 2018
Co-Authors: Jing Hao, Varun Jog
Abstract:

We establish a lower bound on the entropy of weighted sums of (possibly dependent) Random Variables (X 1 , X 2 , …, X n) possessing a symmetric joint distribution. Our lower bound is in terms of the joint entropy of (X 1 , X 2 , …, Xn). We show that for $n$ ≥ 3, the lower bound is tight if and only if X i 'S are i.i.d. Gaussian Random Variables. For $n$ = 2 there are numerous other cases of equality apart from i.i.d. Gaussians, which we completely characterize. Going beyond sums, we also present an inequality for certain linear transformations of (X 1 , …, X n.). Our primary technical contribution lies in the analysis of the equality cases, and our approach relies on the geometry and the symmetry of the problem.

• ##### an entropy inequality for symmetric Random Variables
arXiv: Information Theory, 2018
Co-Authors: Jing Hao, Varun Jog
Abstract:

We establish a lower bound on the entropy of weighted sums of (possibly dependent) Random Variables $(X_1, X_2, \dots, X_n)$ possessing a symmetric joint distribution. Our lower bound is in terms of the joint entropy of $(X_1, X_2, \dots, X_n)$. We show that for $n \geq 3$, the lower bound is tight if and only if $X_i$'s are i.i.d.\ Gaussian Random Variables. For $n=2$ there are numerous other cases of equality apart from i.i.d.\ Gaussians, which we completely characterize. Going beyond sums, we also present an inequality for certain linear transformations of $(X_1, \dots, X_n)$. Our primary technical contribution lies in the analysis of the equality cases, and our approach relies on the geometry and the symmetry of the problem.

### Chaoming Hwang - One of the best experts on this subject based on the ideXlab platform.

• ##### a theorem of renewal process for fuzzy Random Variables and its application
Fuzzy Sets and Systems, 2000
Co-Authors: Chaoming Hwang
Abstract:

The fuzzy set was introduced by Zadeh (1965) and the concept of fuzzy Random Variables was provided by Kwakernaak (1981). Sequences of independent and identical distributed fuzzy Random Variables were considered by Kruse (1982). He also showed the strong law of large numbers for fuzzy Random Variables. In this paper, we consider the stochastic process for fuzzy Random Variables and prove a theorem for fuzzy rate of a fuzzy renewal process.