Dependent Random Variables Research Articles

Complete moment convergence for maximum of randomly weighted sums of martingale difference sequences

In this paper, complete moment convergence for maximum of randomly weighted sums and complete convergence for randomly indexed sums of martingale difference sequences (MDS) are investigated under some proper and sufficient conditions. A Marcinkiewicz–Zygmund type strong law of large numbers (MZSLLN) for MDS is obtained. In addition, relationships among weights, weight functions and boundary functions are revealed in a sense. The results obtained in the paper generalize some corresponding ones for independent and some dependent random variables. As an application, strong consistency for estimators in a nonparametric regression model is established.

On the rate of convergence in the invariance principle for weakly dependent random variables

UDC 519.21 We consider nonstationary sequences of φ -mixing random variables. By using the Levy–Prokhorov distance, we estimate the rate of convergence in the invariance principle for nonstationary φ -mixing random variables. The obtained results extend and generalize several known results for nonstationary φ -mixing random variables.

Complete Convergence for M-Pairwise Negatively Dependent Random Variables

Hsu and Robbins (1947) introduce the concept complete convergence as follows. A sequence of random variables is said to converge completely to a constant if for all The converse is true if the are independent. They also show that the sequence of arithmetic means of independent and identically distributed random variables converges completely to the expected value if the variance of the summands is finite. Erdös proved the converse. The result of Hsu-Robbins-Erdös is a fundamental theorem in probability theory and has been generalized and extended in several directions by many authors. In this paper, let be a sequence of positive constants with and be a sequence of m-pairwise negatively dependent random variables. We study the complete convergence for m-pairwise negatively dependent random variables under mild condition . Our results obtained in the paper generalize the corresponding ones for pairwise independent and identically distributed random variables and also pairwise negatively dependent random variables.

Asymptotic analysis of synchrosqueezing transform—toward statistical inference with nonlinear-type time-frequency analysis

We provide a statistical analysis of a tool in nonlinear-type time-frequency analysis, the synchrosqueezing transform (SST), for both the null and nonnull cases. The intricate nonlinear interaction of different quantities in SST is quantified by carefully analyzing relevant multivariate complex Gaussian random variables. Specifically, we provide the quotient distribution of dependent and improper complex Gaussian random variables. Then a central limit theorem result for SST is established. As an example, we provide a block bootstrap scheme based on the established SST theory to test if a given time series contains oscillatory components.

Convergence of the Kiefer–Wolfowitz algorithm in the presence of discontinuities

AbstractIn this paper we estimate the expected error of a stochastic approximation algorithm where the maximum of a function is found using finite differences of a stochastic representation of that function. An error estimate of the order $n^{-1/5}$ for the nth iteration is achieved using suitable parameters. The novelty with respect to previous studies is that we allow the stochastic representation to be discontinuous and to consist of possibly dependent random variables (satisfying a mixing condition).

The distribution of the sum of two dependent randomly weighted random variables with applications

There has been much work on the distribution of independent or dependent random variables. But we are not aware of any work giving exact results for the distribution of the sum of randomly weighted random variables. In this paper, we derive exact results for the randomly weighted sum of two dependent random variables. The derived expressions are for the cumulative distribution function, conditional expectation, moment generating function, value at risk, expected shortfall and the limiting tail behavior of the randomly weighted sum of two dependent random variables. Two numerical illustrations are given.

An advanced mixed-degree cubature formula for reliability analysis

Efficient assessment of mechanical system reliability subject to arbitrary probability distributions and dependent input parameters signifies an important yet challenging task. To tackle this problem, this study proposes a new moment-based method for reliability analysis of complex mechanical systems which incorporates the advanced mixed-degree cubature formula and vine copula function. To start with, a method integrating the vine copula function and Rosenblatt transformation is developed for transferring the uncertainty with dependent random variables, in which the vine copula is applied for depicting the correlation between random variables. An advanced mixed-degree cubature formula is then established for calculating statistical moments of performance function, which enables to capture sufficient uncertainty information of variables after rotating integral node. The Hermite polynomial model is adopted here to reconstruct the probability distribution of performance function with the statistical moments. To demonstrate the effectiveness of the proposed method, four illustrative examples are implemented in this study, in which Monte Carlo Simulation (MCS) and dimension-reduction techniques, are performed for comparisons. The results show that the proposed method can handle multi-dimensional correlation effectively and well balance computational accuracy and efficiency for both statistical moments and probability distribution evaluations.

Tail Index Estimation of PageRanks in Evolving Random Graphs

Random graphs are subject to the heterogeneities of the distributions of node indices and their dependence structures. Superstar nodes to which a large proportion of nodes attach in the evolving graphs are considered. In the present paper, a statistical analysis of the extremal part of random graphs is considered. We used the extreme value theory regarding sums and maxima of non-stationary random length sequences to evaluate the tail index of the PageRanks and max-linear models of superstar nodes in the evolving graphs where existing nodes or edges can be deleted or not. The evolution is provided by a linear preferential attachment. Our approach is based on the analysis of maxima and sums of the node PageRanks over communities (block maxima and block sums), which can be independent or weakly dependent random variables. By an empirical study, it was found that tail indices of the block maxima and block sums are close to the minimum tail index of representative series extracted from the communities. The tail indices are estimated by data of simulated graphs.

Probability tail for linearly negative quadrant dependent random variables of partial sums and application to linear model

In this paper, we establish a new concentration inequality and complete convergence of weighted sums for arrays of rowwise linearly negative quadrant dependent (LNQD, in short) random variables and obtain a result dealing with complete convergence of first-order autoregressive processes with identically distributed LNQD innovations.

Extrema of the Generalized Allocation Scheme Based on an $m$-Dependent Sequence

We establish some asymptotic estimations for the extrema of the generalized allocation scheme driven by $m$-dependent random variables, which extend some known results of Chuprunov and Fazekash [Discrete Math. Appl., 22 (2012), pp. 307--314] from the independent case to the $m$-dependent case.

Dependent random variables in quantum dynamics

We consider the many-body time evolution of weakly interacting bosons in the mean field regime for initial coherent states. We show that bounded k-particle operators, corresponding to dependent random variables, satisfy both a law of large numbers and a central limit theorem.

On tail behavior of randomly weighted sums of dependent subexponential random variables

In this paper, we revisit the tail behavior of randomly weighted sums and their maxima of dependent subexponential random variables, in which the primary random variables X 1 , … , X n are real-valued and dependent following two general dependence structures, respectively, and the random weights θ 1 , … , θ n are another n positive and arbitrarily dependent random variables, but independent of X 1 , … , X n . . Under some technical conditions, we derive some asymptotic formulas for the tail probability of the randomly weighted sums and their maxima, which coincide with some existing ones in the literature. The merit of our results is that unbounded supports for the random weights are allowed and the distributions of primary random variables can be different.

Complete moment convergence for randomly weighted sums of arrays of rowwise $$m_n$$-extended negatively dependent random variables and its applications

Complete moment convergence for randomly weighted sums of arrays of rowwise $$m_n$$-extended negatively dependent random variables and its applications

On almost sure convergence for double arrays of dependent random variables

In this paper, we investigate some sufficient conditions of Chung type almost sure convergence and strong law of large numbers of double array of arbitrarily dependent random variables. In the proofs, the techniques developed by Ghosal and Chandra are employed.

The chover-type law of iterated logarithm for the weighted sums of negatively superadditive dependent random variables

Let be a sequence of NSD random variables which is stochastically dominated by a random variable X satisfying that X is tailed, Let be an array of real numbers with In this paper, the author proves the Chover-type law of the iterated logarithm for weighted partial sums of NSD random variables, and also gives some classical examples. The results obtained improve and generalize the known results.

Complete moment convergence for randomly weighted sums of negatively superadditive dependent random variables

In this article, the complete moment convergence for randomly weighted sums of negatively superadditive dependent (NSD, for short) random variables is studied. Several sufficient conditions of the complete moment convergence for randomly weighted sums of NSD random variables are presented. The results obtained in the article extend some corresponding ones in the literature.

An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables

There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

NEW TAIL PROBABILITY TYPE INEQUALITIES AND COMPLETE CONVERGENCE FOR WOD RANDOM VARIABLES WITH APPLICATION TO LINEAR MODEL GENERATED BY WOD ERRORS

In this paper, we prove an exponential type inequality for unbounded Widely Orthant Dependent (WOD) random variables. Then we will use this inequality for establishing the almost complete convergence for a sequence of widely dependent random variables (WOD). As an application, we combine Markov inequalities and the new exponential inequality to discuss almost complete convergence of errors generated by a sequence of dependent random variables (WOD) in linear models.

MIAMI: mutual information-based analysis of multiplex imaging data.

MotivationStudying the interaction or co-expression of the proteins or markers in the tumor microenvironment of cancer subjects can be crucial in the assessment of risks, such as death or recurrence. In the conventional approach, the cells need to be declared positive or negative for a marker based on its intensity. For multiple markers, manual thresholds are required for all the markers, which can become cumbersome. The performance of the subsequent analysis relies heavily on this step and thus suffers from subjectivity and lacks robustness.ResultsWe present a new method where different marker intensities are viewed as dependent random variables, and the mutual information (MI) between them is considered to be a metric of co-expression. Estimation of the joint density, as required in the traditional form of MI, becomes increasingly challenging as the number of markers increases. We consider an alternative formulation of MI which is conceptually similar but has an efficient estimation technique for which we develop a new generalization. With the proposed method, we analyzed a lung cancer dataset finding the co-expression of the markers, HLA-DR and CK to be associated with survival. We also analyzed a triple negative breast cancer dataset finding the co-expression of the immuno-regulatory proteins, PD1, PD-L1, Lag3 and IDO, to be associated with disease recurrence. We demonstrated the robustness of our method through different simulation studies.Availability and implementationThe associated R package can be found here, https://github.com/sealx017/MIAMI.Supplementary information Supplementary data are available at Bioinformatics online.

Limit theorems for dependent random variables with infinite means

We investigate necessary and sufficient conditions for the convergence in probability of weighted averages of random variables with infinite means. The obtained results are valid for a wide range of dependence structures, they extend and improve the corresponding theorems of Adler (2012) and Nakata (2016), established in the independent case.