Uniform Integrability Condition Research Articles

On weak laws of large numbers for maximal partial sums of pairwise independent random variables

This paper develops Rio’s method [11] to prove the weak law of large numbers for maximal partial sums of pairwise independent random variables. The method allows us to avoid using the Kolmogorov maximal inequality. As an application, a weak law of large numbers for maximal partial sums of pairwise independent random variables under a uniform integrability condition is also established. The sharpness of the result is illustrated by an example.

Asymptotic optimality of the binomial-exhaustive policy for polling systems with large switchover times

We study an optimal-control problem of polling systems with large switchover times, when a holding cost is incurred on the queues. In particular, we consider a stochastic network with a single server that switches between several buffers (queues) according to a pre-specified order, assuming that the switchover times between the queues are large relative to the processing times of individual jobs. Due to its complexity, computing an optimal control for such a system is prohibitive, and so we instead search for an asymptotically optimal control. To this end, we first solve an optimal control problem for a deterministic relaxation (namely, for a fluid model), that is represented as a hybrid dynamical system. We then “translate” the solution to that fluid problem to a binomial-exhaustive policy for the underlying stochastic system, and prove that this policy is asymptotically optimal in a large-switchover-time scaling regime, provided a certain uniform integrability (UI) condition holds. Finally, we demonstrate that the aforementioned UI condition holds in the following cases: (i) the holding cost has (at most) linear growth, and all service times have finite second moments; (ii) the holding cost grows at most at a polynomial rate (of any degree), and the service-time distributions possess finite moment generating functions.

Analytical study of the first collision time finiteness between two randomly moving particles in a fractured medium

This paper presents the stronger moment conditions which are sufficient to show the finiteness of the first collision time between [Formula: see text]-dimensional Gaussian jump particle and another [Formula: see text]-dimensional Brownian particle in a fractured medium. To obtain these conditions, we use the triangular arrays of the random sequence vector which gives all probable positions of the first collision at any time. This study is an extension of the analytical study which is detailed in Alzulaibani and El-Hadidy [Int. J. Mod. Phys. B 33(28) (2019) 1950334] using the uniform integrability conditions.

Γ-convergence and stochastic homogenization of degenerate integral functionals in weighted Sobolev spaces

We study the$\Gamma$-convergence of nonconvex vectorial integral functionals whose integrands satisfy possibly degenerate growth and coercivity conditions. The latter involve suitable scale-dependent weight functions. We prove that under appropriate uniform integrability conditions on the weight functions, which shall belong to a Muckenhoupt class, the corresponding functionals$\Gamma$-converge, up to subsequences, to a degenerate integral functional defined on a limit weighted Sobolev space. The general analysis is then applied to the case of random stationary integrands and weights to prove a stochastic homogenization result for the corresponding functionals.

Multivariate goodness-of-fit tests based on Wasserstein distance

Goodness-of-fit tests based on the empirical Wasserstein distance are proposed for simple and composite null hypotheses involving general multivariate distributions. For group families, the procedure is to be implemented after preliminary reduction of the data via invariance. This property allows for calculation of exact critical values and p-values at finite sample sizes. Applications include testing for location–scale families and testing for families arising from affine transformations, such as elliptical distributions with given standard radial density and unspecified location vector and scatter matrix. A novel test for multivariate normality with unspecified mean vector and covariance matrix arises as a special case. For more general parametric families, we propose a parametric bootstrap procedure to calculate critical values. The lack of asymptotic distribution theory for the empirical Wasserstein distance means that the validity of the parametric bootstrap under the null hypothesis remains a conjecture. Nevertheless, we show that the test is consistent against fixed alternatives. To this end, we prove a uniform law of large numbers for the empirical distribution in Wasserstein distance, where the uniformity is over any class of underlying distributions satisfying a uniform integrability condition but no additional moment assumptions. The calculation of test statistics boils down to solving the well-studied semi-discrete optimal transport problem. Extensive numerical experiments demonstrate the practical feasibility and the excellent performance of the proposed tests for the Wasserstein distance of order p=1 and p=2 and for dimensions at least up to d=5. The simulations also lend support to the conjecture of the asymptotic validity of the parametric bootstrap.

Analysis of the finiteness for the first collision time between two randomly moving particles

The finiteness of the collision time between two different randomly moving particles is presented by providing more useful analysis that gives stronger and finite moment. The triangular arrays and the uniform integrability conditions of the all probable positions non-stationary random sequence are used. In addition, an important property of Marcinkiewicz laws of large numbers and Hoffman-Jorgensen inequality are presented in this analysis. All of them are deriving to provide the sufficient conditions that give more stronger moments of the first meeting time in the probability space.

Optimal Solutions to Infinite-Player Stochastic Teams and Mean-Field Teams

We study stochastic static teams with countably infinite number of decision makers (DMs), with the goal of obtaining (globally) optimal policies under a decentralized information structure. We present sufficient conditions to connect the concepts of team optimality and person-by-person optimality for static teams with countably infinite number of DMs. We show that under uniform integrability and uniform convergence conditions, an optimal policy for static teams with countably infinite number of DMs can be established as the limit of sequences of optimal policies for static teams with $N$ DMs as $N \to \infty$ . Under the presence of a symmetry condition, we relax the conditions and this leads to optimal results for a large class of mean-field optimal team problems where the existing results have been limited to person-by-person optimality and not global optimality (under strict decentralization). In particular, we establish the optimality of symmetric (i.e., identical) policies for such problems. As a further condition, this optimality result leads to an existence result for mean-field teams. We consider a number of illustrative examples where the theory is applied to setups with either infinitely many DMs or an infinite-horizon stochastic control problem reduced to a static team.

The finiteness of the first meeting time between a Gaussian jump and a Brownian particles in the fluid

In this paper, we present more analysis about the finiteness of the first meeting time between Gaussian jump and Brownian particles in the fluid. Rather than using the uniform integrability conditions which are detailed in El-Hadidy [Mod. Phys. Lett. B 33(22) (2019) 1950256], we show the triangular arrays of the random sequence that represents the probable positions of the first meeting at any time [Formula: see text], to get the stronger moment conditions which are sufficient for the finiteness of this time.

Fatou's Lemma for Weakly Converging Measures under the Uniform Integrability Condition

This paper describes Fatou's lemma for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to the...

Supplemental Material to 'Self-Enforcing Debt and Rational Bubbles'

In this document, we present two technical results. We first provide a formal proof that optimal consumption is strictly positive even if self-enforcing and not-too-tight debt limits typically bind. We also propose a simple example to illustrate that the uniform integrability condition imposed by Bidian and Bejan (2015) on the discounted debt limits is not necessarily satisfied at equilibrium.

Mean Convergence Theorems and Weak Laws of Large Numbers for Arrays of Measurable Operators under Some Conditions of Uniform Integrability

In this paper, we introduce the notions of uniform integrability in the Cesaro sense, h-integrability with respect to the array of constants {ani}, and h-integrability with exponent r for an array of measurable operators. Then, we establish some mean convergence theorems and weak laws of large numbers for arrays of measurable operators under some conditions related to these notions.

Minimal Root’s embeddings for general starting and target distributions

Most results regarding Skorokhod embedding problems SEP so far rely on the assumption that the corresponding stopped process is uniformly integrable, which is equivalent to the convex ordering condition Uμ≤Uν when the underlying process is a local martingale. In this paper, we study the existence, construction of Root’s solutions to SEP, in the absence of this convex ordering condition. We replace the uniform integrability condition by the minimality condition (Monroe, 1972), as the criterion of “good” solutions. A sufficient and necessary condition (in terms of local time) for minimality is given. We also discuss the optimality of such minimal solutions. These results extend the generality of the results given by Cox and Wang (2013) and Gassiat et al. (2015). At last, we extend all the results above to multi-marginal embedding problems based on the work of Cox et al. (2018).

Some convergence properties for the maximum of partial sums of m-negatively associated random variables

In this paper, we present the Spitzer type law of large numbers for the maximum of partial sums of m-negatively associated random variables $$\left\{ X_n,n\ge 1\right\} $$ and the $$L_p$$ convergence property under the Cesaro uniform integrability condition for the maximum of partial sums. In addition, we give some simulations to verify the convergence behavior of $$\frac{1}{\psi (n)}\sum _{i=1}^{n}X_i$$ which is in accordance with our theoretical result. The main results obtained in this paper extend and improve the corresponding ones for negatively associated random variables.

Limiting behaviour for arrays of rowwise widely orthant dependent random variables under conditions of R − h-integrability and its applications

ABSTRACTIn this paper, we mainly study the convergence, complete convergence and complete moment convergence for arrays of rowwise widely orthant dependent (WOD, for short) random variables under some conditions of R-h-integrability. The results in this paper generalize and improve the corresponding ones of Sung et al., Weak laws of large numbers for arrays under a condition of uniform integrability, Journal of the Korean Mathematical Society 45 (2008), pp. 289–300, Wang and Hu, Weak laws of large numbers for arrays of dependent random variables, Stochastics: An International Journal of Probability and Stochastic Processes 86 (2014), pp. 759–775, and Wu et al., Limiting behaviour for arrays of rowwise END random variables under conditions of h-integrability, Stochastics: An International Journal of Probability and Stochastic Processes 87 (2014), pp. 409–423. As applications of the complete convergence that we established, we present some results on complete consistency for the estimator in a nonparametric regression model based on WOD errors.

Fatou's lemma for weakly converging measures under the uniform integrability condition

В статье устанавливаются лемма Фату и теорема Лебега о мажорируемой сходимости для слабо сходящихся последовательностей конечных мер и равномерно интегрируемых по этим мерам функций. Заметка содержит также новые формулировки равномерной леммы Фату, равномерной теоремы Лебега о мажорируемой сходимости, теоремы Данфорда-Петтиса и фундаментальной теоремы о мерах Янга, основанные на эквивалентности условия равномерной интегрируемости и кажущегося более слабым условия асимптотической равномерной интегрируемости для последовательностей функций и конечных мер.

The Gamma Stein equation and noncentral de Jong theorems

We study the Stein equation associated with the one-dimensional Gamma distribution, and provide novel bounds, allowing one to effectively deal with test functions supported by the whole real line. We apply our estimates to derive new quantitative results involving random variables that are non-linear functionals of random fields, namely: (i) a non-central quantitative de Jong theorem for sequences of degenerate $U$-statistics satisfying minimal uniform integrability conditions, significantly extending previous findings by de Jong (J. Multivariate Anal. 34 (1990) 275–289), Nourdin, Peccati and Reinert (Ann. Probab. 38 (2010) 1947–1985) and Dobler and Peccati (Electron. J. Probab. 22 (2017) no. 2), (ii) a new Gamma approximation bound on the Poisson space, refining previous estimates by Peccati and Thale (ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 525–560) and (iii) new Gamma bounds on a Gaussian space, strengthening estimates by Nourdin and Peccati (Probab. Theory Related Fields 145 (2009) 75–118). As a by-product of our analysis, we also deduce a new inequality for Gamma approximations via exchangeable pairs, that is of independent interest.

Some convergence properties for partial sums of widely orthant dependent random variables and their statistical applications

In this paper, we present the $$L_p$$ convergence for partial sums $$S_n=\sum _{k=1}^nX_k$$ under the Cesaro uniform integrability condition and the complete convergence for the maximum of $$S_n$$ for sequences of widely orthant dependent random variables $$\{X_n,n\ge 1\}.$$ Some of the results extend the corresponding ones in reference. As applications, we get the complete consistency and the strong consistency for the estimator in a nonparametric regression model.

On Balder’s Existence Theorem for Infinite-Horizon Optimal Control Problems

Balder’s well-known existence theorem (1983) for infinite-horizon optimal control problems is extended to the case in which the integral functional is understood as an improper integral. Simultaneously, the condition of strong uniform integrability (over all admissible controls and trajectories) of the positive part max{f0, 0} of the utility function (integrand) f0 is relaxed to the requirement that the integrals of f0 over intervals [T, T′] be uniformly bounded above by a function ω(T, T′) such that ω(T, T′) → 0 as T, T′→∞. This requirement was proposed by A.V. Dmitruk and N.V. Kuz’kina (2005); however, the proof in the present paper does not follow their scheme, but is instead derived in a rather simple way from the auxiliary results of Balder himself. An illustrative example is also given.

Weak and strong laws of large numbers for arrays of rowwise END random variables and their applications

In the paper, the Marcinkiewicz–Zygmund type moment inequality for extended negatively dependent (END, in short) random variables is established. Under some suitable conditions of uniform integrability, the $$L_r$$ convergence, weak law of large numbers and strong law of large numbers for usual normed sums and weighted sums of arrays of rowwise END random variables are investigated by using the Marcinkiewicz–Zygmund type moment inequality. In addition, some applications of the $$L_r$$ convergence, weak and strong laws of large numbers to nonparametric regression models based on END errors are provided. The results obtained in the paper generalize or improve some corresponding ones for negatively associated random variables and negatively orthant dependent random variables.

Some Kinds of Uniform Integrability and Laws of Large Numbers in Noncommutative Probability

The aim of this paper is to propose some equivalent conditions of uniform integrability for a sequence of measurable operators. Based on this result, we introduce some kinds of Cesaro uniform integrability in noncommutative probability. In addition, several results on the laws of large numbers related to these notions are established. Our results not only generalize related previously reported results but also improve them.