Dependent Random Variables Research Articles

On the Tail Behavior for Randomly Weighted Sums of Dependent Random Variables with its Applications to Risk Measures

On the Tail Behavior for Randomly Weighted Sums of Dependent Random Variables with its Applications to Risk Measures

The heavy-tail behavior of the difference of two dependent random variables

The heavy-tail behavior of the difference of two dependent random variables

Selected Reinsurance Models

In this paper, we investigate non-classical reinsurance models. Two kinds of such models are presented. One is based on dependent binomial distributions and the second on fuzzy numbers. First, we study dependent random variables representing claims using copulas. We investigate the number of claims the reinsurer covers and the total value of covered claims. We present the influence of the degree of dependence and different copulas on the number and the value of the claims covered by the reinsurer. Second, we analyze the case in which the main parameter of the model, the probability that the reinsurer covers the claim, is uncertain. We treat such a parameter as a fuzzy number in this case and combine randomness and fuzziness. We also study the case when the parameter of a copula which describes the degree of dependence is uncertain.

The weak law of large numbers for weighted sums of m-asymptotic negatively associated random variables

In this article, we mainly investigate the Kolmogorov-Feller type weak law of large numbers for weighted sums of m-asymptotic negatively associated (m-ANA, for short) random variables. Some sufficient conditions to prove the Kolmogorov-Feller type weak law of large numbers are provided. The results obtained in this article extend the corresponding ones of Naderi, Boukhari, and Matula (2022) for negatively superadditive dependent (NSD, for short) random variables.

Weak laws of large numbers for weighted sums of arrays of random variables under sublinear expectation

. In this article, we introduce a new concept of integrability called extended residually h-integrability with exponent r, which deals with weighted sums of arrays under sublinear expectation. The classical copy of this concept extends the residually h-integrability given by Wang, Hu, and Volodin (2018). On this basis, we obtain three forms of weak laws of large numbers for weighted sums of arrays of rowwise negatively dependent or rowwise independent random variables, respectively, under sublinear expectation. As corollaries, we get some weak laws of large numbers for arrays of random variables with normalizing factor k n under sublinear expectation.

Generalized likelihood ratio method for stochastic models with uniform random numbers as inputs

We propose a new unbiased stochastic gradient estimator for a family of stochastic models driven by uniform random numbers as inputs. Dropping the requirement that the tails of the density of the input random variables decay smoothly, the estimator extends the applicability of the generalized likelihood ratio (GLR) method. We demonstrate the new estimator for several general classes of input random variates, including independent inverse transform random variates and dependent input random variables governed by an Archimedean copula. We show how the new estimator works in settings such as density estimation, and we illustrate applications to credit risk derivatives. Numerical experiments substantiate broad applicability and flexibility in dealing with discontinuities in the sample performance.

Three-parameter approximations of sums of locally dependent random variables via Stein’s method

Three-parameter approximations of sums of locally dependent random variables via Stein’s method

The convergence properties for randomly weighted sums of widely negative dependent random variables under sub-linear expectations with related statistical applications

In this paper, we study the complete convergence and complete moment convergence for randomly weighted sums of arrays of rowwise widely negative dependent random variables in sub-linear expectation space under some appropriate conditions, which extend and improve the corresponding ones in classical probability space to the case of sub-linear expectation space. And we obtain a strong law of large numbers for the randomly weighted sums of arrays of rowwise widely negative dependent random variables. As applications of our main results, we not only present a result on the complete consistency for the weighted estimator in a nonparametric regression model, but also obtain the complete consistency for the least squares estimators in errors-in-variables regression models based on widely negative dependent errors under sub-linear expectations. We perform some numerical simulations to verify the validity of the theoretical results and a real example is analysed for illustration.

Stochastic comparisons of second largest order statistics with dependent heterogeneous random variables

In the context of actuarial science, the second largest claim amount is crucial to insurance analysis since they provide useful information for determining annual premium. In this article, we provide sufficient conditions of the second largest claim amounts arising from two sets of dependent and heterogeneous individual risk models according to various stochastic orders. It is first shown that the reversed hazard order between the occurrence probabilities vectors implies the reversed hazard order of the second largest claim amounts under certain conditions. Second, sufficient conditions are established for the usual stochastic ordering of the second largest claim amounts arising from heterogeneous dependent individual risk models under copula dependence. Finally, several examples illustrating the theoretical results are presented as well.

Strong laws for weighted sums of widely orthant dependent random variables and applications

Abstract In this study, the strong law of large numbers and the convergence rate for weighted sums of non-identically distributed widely orthant dependent random variables are established. As applications, the strong consistency for weighted estimator in nonparametric regression model and the rate of strong consistency for least-squares estimator in multiple linear regression model are obtained. Some numerical simulations are also provided to verify the validity of the theoretical results.

Berry-Esseen bounds for self-normalized sums of locally dependent random variables

Berry-Esseen bounds for self-normalized sums of locally dependent random variables

Strong Convergence Properties for Weighted Sums of Extended Negatively Dependent Random Variables Under Sub-linear Expectations with Statistical Applications

Strong Convergence Properties for Weighted Sums of Extended Negatively Dependent Random Variables Under Sub-linear Expectations with Statistical Applications

Moderate deviations for multidimensional aggregate claims with arbitrary dependence between claim sizes and waiting times

Consider a multidimensional renewal risk model, in which the claim sizes X k , k ≥ 1 , form a sequence of independent and identically distributed random vectors with non negative components allowed to be dependent on each other, and the inter-arrival times form a sequence of p-extended negatively dependent random variables. By allowing arbitrary dependence between claim sizes and waiting times, we obtain the moderate deviations for the multidimensional aggregate claims with consistently varying tails.

Large Deviations for the Ground State of Weakly Interacting Bose Gases

AbstractWe consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose–Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables due to the bosons’ correlation. We prove that in the limit $$N \rightarrow \infty $$ N → ∞ bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.

Complete Convergence for Weighted Sums of Pairwise Negative Quadrant Dependent Random Variables

In this work, we developed Jajte’s technique of the strong law of large numbers to the complete convergence for randomly weighted sums of pairwise negative quadrant dependent random variables.

Complete convergence theorems for weighted sums of extended negatively dependent random variables under sub-linear expectations

. This article aims to study and establish the Lai law for the weighted sum of extended negatively dependent random variables under sub-linear expectations. Using inequalities under sub-linear expectation spaces, we extend the complete convergence of weighted sums from classical probability space to sub-linear expectation space.

Multivariate Universal Local Linear Kernel Estimators in Nonparametric Regression: Uniform Consistency

In this paper, for a wide class of nonparametric regression models, new local linear kernel estimators are proposed that are uniformly consistent under close-to-minimal and visual conditions on design points. These estimators are universal in the sense that their designs can be either fixed and not necessarily satisfying the traditional regularity conditions, or random, while not necessarily consisting of independent or weakly dependent random variables. With regard to the design elements, only dense filling of the regression function domain with the design points without any specification of their correlation is assumed. This study extends the dense data methodology and main results of the authors’ previous work for the case of regression functions of several variables.

Статистичні тести для перевірки незалежності випадкових величин, що описують генерацію послідовностей в криптоалгоритмах

When a crypto-primitive, whose functions include the generation of a random/pseudo-random gamma, is admitted to operation, a necessary part of its quality checking is a sta-tistical testing of its output gamma and, often, intermediate gamma(s). Such requirement is applied, for example, to random/pseudo-random number generators (RNG/PRNG), stream ciphers, and block ciphers in different "stream" modes (such as OFB, CBC, etc). There exist widely used and well-known tools for checking the statistical properties of se-quences and generators, which are based on a set of statistical tests, like NIST STS set, Diehard, etc. At the same time, the other and very similar question, about independence of the se-quences (more precisely — independence of the corresponding random variables, that the considered sequences are their realizations) generated in such cryptoalgorithms, usually doesn’t attract enough attention. Nevertheless, it is also of great importance, because the dependence of the sequences can lead to predictability of the output gamma, which makes the cryptoprimitive vulnerable to statistical attacks. Therefore, there are no adequate and suitable tools for checking independence of different sequences, generated in the algo-rithms. In this work we deve¬loped and justified new set of three statistical tests for check-ing independence of random va¬riables, which realizations are internal or output sequences in encryption algorithms or RNG/PRNG. We also calculated reference values for limit sta-tistics for different parameters of sequences and different significance levels of tests. Re-sults of tests applications for independent and dependent random variables are given, which confirm correctness of proposed tests.

Nonparametric estimation of the relative error regression for twice censored and dependent data

Let ( Y i ) 1 ≤ i ≤ n be a sequence of dependent random variables (r.v.) of interest distributed as Y and ( X i ) 1 ≤ i ≤ n be a corresponding d−dimensional vector of covariates taking values on ℝ d . The r.v. Y is twice censored and satisfies the α−mixing property. In this article, we are interested in the relative error regression estimation in the case of twice-censored data under strong mixing conditions. Under suitable assumptions, the estimator’s strong uniform almost sure consistency with rate and asymptotic normality are established. We have highlighted the covariance term, which is relatively uncommon in this context. Furthermore, we give a result about the α-mixing rate in the case of twice-censoring. As far as we know, there is no analogous result with proof in the context of twice-censoring. A simulation study is conducted in both one and two dimensions for the covariate to highlight the performance of our estimator.

Precise large deviations for sums of dependent random variables with subexponential distribution

. In this article, we study the precise large deviations for sums of non identically distributed random variables { X i , i ≥ 1 } with distribution functions { F i , i ≥ 1 } that are widely upper orthant dependent (WUOD, in short) and ϕ-mixing. Moreover, the random variables are dependent, according to some copulas. We assume that the average of the right tails of distribution functions F i is equivalent to the subexponential distribution F.