A note on the almost sure approximation of weakly dependent 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.

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.

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.

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.

Previous article Next article On a Condition of Almost Markov RegularityV. A. StatulyavichusV. A. Statulyavichushttps://doi.org/10.1137/1128030PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] B. Ryauba, The central limit theorem for sums of series of weakly dependent random variables, Litovsk. Mat. Sb., 2 (1962), 193–205, (In Russian.) 29:6526 Google Scholar[2] I. G. Zhurbenko, On the estimation of mixed cumulants for a class of random processes, Theory Prob. Appl., 15 (1970), 525–528 LinkGoogle Scholar[3] I. G. Zhurbenko, On strong estimates of mixed cumulants of random processes, Siberian Math. J., 13 (1972), 302–303 CrossrefGoogle Scholar[4] I. G. Zhurbenko and , N. M. Zuev, The higher spectral densities of stationary processes with mixing, Ukrain. Mat. Ž., 27 (1975), 452–463, 572 53:14620 0325.60039 Google Scholar[5] I. G. Zhurbenko, Mixing conditions and asymptotic theory of stationary time seriesStochastic processes and related topics (Proc. Summer Res. Inst. on Statist. Inference for Stochastic Processes, Indiana Univ., Bloomington, Ind., 1974, Vol. 1; dedicated to Jerzy Neyman), Academic Press, New York, 1975, 259–265 51:9177 0349.60031 Google Scholar[6] V. A. Statulyavichus, Limit theorems for random functions. I, Litovsk. Mat. Sb., 10 (1970), 583–592, (In Russian.) 43:2765 Google Scholar[7] V. A. Statulyavichus, Limit theorems for dependent random variables under various regularity conditions, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, 173–181 55:1420 Google Scholar[8] V. A. Statulyavichus, Application of semi-invariants to asymptotic analysis of distributions of random processesMultivariate analysis, IV (Proc. Fourth Internat. Sympos., Dayton, Ohio, 1975), North-Holland, Amsterdam, 1977, 325–337 57:1614 Google Scholar[9] R. L. Dobrushin, The description of a random field by means of conditional probabilities and conditions for its regularity, Theory Prob. Appl., 13 (1968), 197–224 LinkGoogle Scholar[10] I. A. Ibragimov, Some limit theorems for stationary processes, Theory Prob. Appl., 7 (1962), 349–382 LinkGoogle Scholar[11] M. Rosenblatt, Some remarks on a mixing condition, Ann. Probab., 7 (1979), 170–172 80k:60071a 0925.60052 CrossrefGoogle Scholar[12] V. A. Statulyavichus, Large-deviation theorems for sums of dependent random variables, Lithuanian Math. J., 19 (1979), 289–295 CrossrefGoogle Scholar[13] A. V. Bulinskii, Limit Theorems for Random Processes and Fields, MGU, Moscow, 1981, (In Russian.) Google Scholar[14] V. Statulyavichus, On limit theorems for dependent random variables, Abstracts of Communications of the Third International Vilnius Conference on Probability Theory and Mathematical Statistics, Vilnius: In-t matem. i kibern. AN Lit SSR, 1981, 320–323 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails An Almost-Markov-Type Mixing Condition and Large Deviations for Boolean Models on the LineActa Applicandae Mathematicae, Vol. 96, No. 1-3 | 23 March 2007 Cross Ref Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal DistributionLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Limit Theorems on Large DeviationsLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Every “lower psi-mixing” Markov chain is “interlaced rho-mixing”Stochastic Processes and their Applications, Vol. 72, No. 2 | 1 Dec 1997 Cross Ref Random fields: Applications in cell biologyStochastic Spatial Processes | 16 September 2006 Cross Ref Volume 28, Issue 2| 1984Theory of Probability & Its Applications219-469 History Submitted:10 January 1983Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1128030Article page range:pp. 379-383ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Problem statement: When analyzing random variables it was useful to measure the degree of their monotone dependence or compare pairs of random variables with respect to their monotonicity. Existing coefficients measure general or linear dependence of random variables. Developing a measure of monotonicity was useful for practical applications as well as for general theory, since monotonicity was an important type of dependence. Approach: Existing measures of dependence are briefly reviewed. The Reimann coefficient was generalized to arbitrary random variables with finite variances. Results: The article describes criteria for monotone dependence of two random variables and introduces a measure of this dependence-monotonicity coefficient. The advantages of this coefficient are shown in comparison with other global measures of dependence. It was shown that the monotonicity coefficient satisfies natural conditions for a monotonicity measure and that it had properties similar to the properties of the Pearson correlation; in particular, it equals 1 (-1) if and only if the pair X, Y was comonotonic (counter-monotonic). The monotonicity coefficient was calculated for some bivariate distributions and the sample version of the coefficient was defined. Conclusion/Recommendations: The monotonicity coefficient should be used to compare pairs of random variables (such as returns from financial assets) with respect to their degree of monotone dependence. In the problems where the monotone relation of two variables has a random noise, the monotonicity coefficient can be used to estimate variance and other central moments of the noise. By calculating the sample version of the coefficient one will quickly find pairs of monotone dependent variables in a big dataset."

Let be an array of rowwise extended negatively dependent (END) random variables. The complete convergence for weighted sums of arrays of rowwise END random variables is studied, which complements and improves the corresponding result of Baek and Park [2] and Baek et al. [1]. As an application, the Baum–Katz type result for arrays of rowwise END random variables is obtained.

In this article, the complete convergence for the maximum of weighted sums of widely negative orthant dependent (WNOD, in short) random variables is investigated under some suitable moment conditions. Some sufficient conditions to prove the complete convergence are provided. The results obtained in the paper generalize some corresponding ones for some dependent random variables. As an application, the complete consistency for the weighted estimator in a nonparametric regression model is established. Finally, we present some simulations to show the consistency for the nearest neighbour weight function estimator in a nonparametric regression model.

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

Let {(Xi,Yi)}i=1n be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cramér-type moderate deviation theorem for the general self-normalized sum ∑ i=1nXi/(∑i=1nYi2)1/2, which unifies and extends the classical Cramér (Actual. Sci. Ind. 736 (1938) 5–23) theorem and the self-normalized Cramér-type moderate deviation theorems by Jing, Shao and Wang (Ann. Probab. 31 (2003) 2167–2215) as well as the further refined version by Wang (J. Theoret. Probab. 24 (2011) 307–329). The advantage of our result is evidenced through successful applications to weakly dependent random variables and self-normalized winsorized mean. Specifically, by applying our new framework on general self-normalized sum, we significantly improve Cramér-type moderate deviation theorems for one-dependent random variables, geometrically β-mixing random variables and causal processes under geometrical moment contraction. As an additional application, we also derive the Cramér-type moderate deviation theorems for self-normalized winsorized mean.

Direct computation of the design point of a stochastic structure using a finite element code

In this paper, we consolidate into one two separate problems - dependent random variables with independent subsets and construction of a joint distribution with given marginals. Let N = {1,2,3,...} and X = {Xn; n ∊ N} be a sequence of random variables with nondegenerate one-dimensional marginal distributions {Fn; n ∊ N}. An example is constructed to show that there exists a sequence of random variables Y = {Yn; n ∊ N} such that the components of a subset of Y are independent if and only if its size is ≦ k, where k ≧ 2 is a prefixed integer. Furthermore, the one-dimensional marginal distributions of Y are those of X.

Complete convergence and complete moment convergence for maximal weighted sums of arrays of rowwise extended negatively dependent random variables with statistical applications

The concept of \(m\)-extended negatively dependent (\(m\)-END, in short) random variables is introduced and the Kolmogorov exponential inequality for \(m\)-END random variables is established. As applications of the Kolmogorov exponential inequality, we further investigate the complete convergence for arrays of rowwise \(m\)-END random variables and the complete consistency for the estimator of nonparametric regression models based on \(m\)-END errors. Our results generalize and improve some known ones for independent random variables and dependent random variables.