Dependent Random Variables with Independent Subsets - II

Previous article Next article The Central Limit Theorem for a Sequence of Random Variables with a Slowly Growing Number of DependencesM. B. Petrovskaya and A. M. LeontovichM. B. Petrovskaya and A. M. Leontovichhttps://doi.org/10.1137/1127089PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. B. Ivanov, Cell Foundations of Plant Growth, Nauka, Moscow, 1974, (In Russian.) Google Scholar[2] Z. A. Lomnicki and , S. K. Zaremba, A further instance of the central limit theorem for dependent random variables, Math. Z, 66 (1957), 490–494 20:6152 0077.33101 CrossrefGoogle Scholar[3] A. F. Ronzhin, On limit theorems for sums of dependent random variables, Theory Prob. Appl., 24 (1979), 549–559 0435.60023 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails When Janson meets McDiarmid: Bounded difference inequalities under graph-dependenceStatistics & Probability Letters, Vol. 17 Cross Ref Weighted dependency graphsElectronic Journal of Probability, Vol. 23, No. none Cross Ref Dependency graphs and mod-Gaussian convergence7 December 2016 Cross Ref A law of large numbers for nearest neighbour statistics12 August 2008 | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 464, No. 2100 Cross Ref Central limit theorems for k-nearest neighbour distancesStochastic Processes and their Applications, Vol. 85, No. 2 Cross Ref On the variance of the random sphere of influence graphRandom Structures and Algorithms, Vol. 14, No. 2 Cross Ref On a Theorem of JansonV. G. Mikhailov17 July 2006 | Theory of Probability & Its Applications, Vol. 36, No. 1AbstractPDF (378 KB)Asymptotic normality of some Graph-Related statistics14 July 2016 | Journal of Applied Probability, Vol. 26, No. 1 Cross Ref On a Condition for Slowly Growing Number of DependenciesA. M. Leontovich28 July 2006 | Theory of Probability & Its Applications, Vol. 30, No. 1AbstractPDF (623 KB) Volume 27, Issue 4| 1983Theory of Probability & Its Applications History Submitted:20 December 1977Published online:17 July 2006 InformationCopyright © 1983 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1127089Article page range:pp. 815-825ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

We study the class of random entire functions given by power series, in which the coefficients are formed as the product of an arbitrary sequence of complex numbers and two sequences of random variables. One of them is the Rademacher sequence, and the other is an arbitrary complex-valued sequence from the class of sequences of random variables, determined by a certain restriction on the growth of absolute moments of a fixed degree from the maximum of the module of each finite subset of random variables. In the paper we prove sharp Wiman–Valiron’s type inequality for such random entire functions, which for given p∈(0;1) holds with a probability p outside some set of finite logarithmic measure. We also considered another class of random entire functions given by power series with coefficients, which, as above, are pairwise products of the elements of an arbitrary sequence of complex numbers and a sequence of complex-valued random variables described above. In this case, similar new statements about not improvable inequalities are also obtained.

Let ${X_n;n\geq 0}$ be a sequence of random variables. We consider its geometrically weighted series $\xi(\beta)=\sum_{n=0}^\infty \betaX_n$ for $0<\beta < 1$. This paper proves that $\xi (\beta)$ can be approximated by $\sum_{n=0}^\infty \beta^n Y_n$ under some suitable conditions, where ${Y_n; n \geq 0}$ is a sequence of independent normal random variables. Applications to the law of the iterated logarithm for $\xi(\beta)$ are also discussed.

A sequence of random variables, $Y_0,Y_1,Y_2,\ldots\,$, is called standard if there exists a one-sided isomorphism between it and a sequence of independent random variables. In this paper it is demonstrated that the sequence arising from the past of the $T,T^{-1}$ map is not standard.

In this paper, the complete convergence for maximum of weighted sums of widely negative orthant dependent (WNOD) random variables are investigated. Some sufficient conditions for the convergence are provided and a relationship between the weight and the boundary function is revealed. Additionally, a Marcinkiewicz-Zygmund type strong law of large number for weighted sums of WNOD random variables is obtained. The results obtained in this paper generalize some corresponding ones for independent and some dependent random variables. As an application, the strong consistency for the weighted estimator in a non-parametric regression model is established. MR(2010) Subject Classification: 60F15; 62G05.

The stress-strength models have been intensively investigated in the literature in regards of estimating the reliability θ = P (X > Y) using parametric and nonparametric approaches under different sampling schemes when X and Y are independent random variables. In this paper, we consider the problem of estimating θ when (X, Y) are dependent random variables with a bivariate underlying distribution. The empirical and kernel estimates of θ = P (X > Y), based on bivariate ranked set sampling (BVRSS) are considered, when (X, Y) are paired dependent continuous random variables. The estimators obtained are compared to their counterpart, bivariate simple random sampling (BVSRS), via the bias and mean square error (MSE). We demonstrate that the suggested estimators based on BVRSS are more efficient than those based on BVSRS. A simulation study is conducted to gain insight into the performance of the proposed estimators. A real data example is provided to illustrate the process.

CHAPTER 5 - SUBSEQUENCES OF SEQUENCES OF RANDOM VARIABLES

Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent or independent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. The approaches developed in this paper are flexible and have a wide range of applications for both symmetric and non-symmetric distributions and also for both skewed and non-skewed copulas with absolutely continuous random variables that could contain a negative range, for instance, generalized skewed-t distribution and skewed-t Copulas. Thus, our findings are useful for academics, practitioners, and policy makers.

General results on the applicability of the strong law of large numbers to a sequence of dependent random variables, as formulated in terms of estimates for the moments of sums of such variables, are applied to give new conditions of the applicability of this law to (in a wide sense) a stationary sequence of random variables.

An elementary proof of de Finetti’s theorem

Cover and Zhang proved a certain reversal of the Entropy Power Inequality for the sum of (possibly dependent) random variables possessing the same log-concave density, and what is more that log-concave densities were the only densities that satisfied such an inequality. In this work the authors consider the analogous reversal of recent Renyi Entropy Power Inequalities for random vectors and again show that not only do they hold for s-concave densities, but that s-concave densities are characterized by satisfying said inequalities.

Estimation of parameters in time and rate dependent materials is frequently performed by least squares regression. Time and rate dependent material models are represented mathematically by systems of differential equations. As predictions of dynamical systems are history dependent, the residuals produced by regression estimators are dependent random variables and the estimates may therefore be biased. A new approach using time series analysis methods, which offers a way to circumvent the problems related to dependent residuals, is presented. In the new approach, the residuals are obtained via a continuous-discrete time Kalman filter. A simulation study, with models suitable for both high temperature creep and transient loading, has been performed. A comparison between the proposed methods and standard least squares regression shows that the model parameters are estimated with up to 40% higher precision. An analysis of the residuals clearly shows that the residuals of the newestimator form an independent sequence of random variables. The results also indicate that the proposed estimator remains unbiased whereas the regression estimator may be biased when the excitation increases.

As we have seen, a sequence of random variables can describe how some process evolves in time. For example, the gambler’s accumulated winnings at time n, that is, after the nth play of a game of chance, is given by the random variable Sn = Xl +...Xn, where the X’s are independent random variables representing the gambler’s winnings at each play of the game. For convenience, we can think of the subscript n of any such sequence of random variables as denoting the nth moment of time, and the value of the random variable Sn as the measurement made at that moment. The sequence therefore represents a “picture” of the evolution of the process at the times 1, 2 ,... .

CHAPTER 6 - ESTIMATION OF PARAMETERS OF DISTRIBUTIONS

Copulas have become an important analytic tool for characterizing multivariate distributions and dependence. One is often interested in simulating data from copula estimates. The process can be analytically and computationally complex and usually involves steps that are unique to a given parametric copula. We describe an alternative approach that uses \probability{proportional{to{size" (PPS) random sampling with weights formed from the copula likelihood function. The method is flexible and can be applied to parametric and nonparametric marginal density estimates. The precision of the simulation can be calibrated by adjusting the density of the multidimensional grid used in the simulation process. The approach is fully transparent to any copula function with continuous random variables. An example evaluates a number of goodness-of-fit criteria and provides strong support for the validity and practicality of the method.