Given a probability space (Ω,, P) and a closed subset X of a Banach lattice, we consider functions f: X × Ω → X and their iterates defined by f¹(x,ω) = f(x,ω₁), , and obtain theorems on the convergence (a.s. and in L¹) of the sequence (fⁿ(x,·)).
In this paper the authors study the convergence properties for arrays of rowwise pairwise negatively quadrant dependent random variables. The results extend and improve the corresponding theorems of T. C. Hu, R. L. Taylor: On the strong law for arrays and for the bootstrap mean and variance, Int. J. Math. Math. Sci 20 (1997), 375–382.
We prove some inequalities for the difference between a joint distribution and the product of its marginals for arbitrary absolutely continuous random variables. Some applications of the obtained inequalities are also presented.
The asymptotic behavior of global errors of functional estimates plays a key role in hypothesis testing and confidence interval building. Whereas for pointwise errors asymptotic normality often easily follows from standard Central Limit Theorems, global errors asymptotics involve some additional techniques such as strong approximation, martingale theory and Poissonization. We review these techniques in the framework of density estimation from independent identically distributed random variables,...
We consider signed Radon random measures on a separable, complete and locally compact metric space and study mean quadratic convergence with respect to vague topology on the space of measures. We prove sufficient conditions in order to obtain mean quadratic convergence. These results are based on some identification properties of signed Radon measures on the product space, also proved in this paper.
We propose two methods to solve multistage stochastic programs when only a (large) finite set of scenarios is available. The usual scenario tree construction to represent non-anticipativity constraints is replaced by alternative discretization schemes coming from non-parametric estimation ideas. In the first method, a penalty term is added to the objective so as to enforce the closeness between decision variables and the Nadaraya–Watson estimation of their conditional expectation. A numerical application...
We give a proof of convergence of moments in the Central Limit Theorem (under the Lyapunov-Lindeberg condition) for triangular arrays, yielding a new estimate of the speed of convergence expressed in terms of νth moments. We also give an application to the convergence in the mean of the pth moments of certain random trigonometric polynomials built from triangular arrays of independent random variables, thereby extending some recent work of Borwein and Lockhart.
Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and bounds on the ensemble then give convergence.
We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion...
We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion...
If is a sequence of real numbers which is good for the ergodic theorem, is the sequence of the integer parts good for the ergodic theorem ? The answer is negative for the mean ergodic theorem and affirmative for the pointwise ergodic theorem.