On the convergence of bilinear and quadratic forms in independent random variables
On the convergence of extreme distributions under power normalization
This paper deals with the convergence in distribution of the maximum of n independent and identically distributed random variables under power normalization. We measure the difference between the actual and asymptotic distributions in terms of the double-log scale. The error committed when replacing the actual distribution of the maximum under power normalization by its asymptotic distribution is studied, assuming that the cumulative distribution function of the random variables is known. Finally,...
On the convergence of moments in the almost sure central limit theorem for stochastic approximation algorithms
We study the almost sure asymptotic behaviour of stochastic approximation algorithms for the search of zero of a real function. The quadratic strong law of large numbers is extended to the powers greater than one. In other words, the convergence of moments in the almost sure central limit theorem (ASCLT) is established. As a by-product of this convergence, one gets another proof of ASCLT for stochastic approximation algorithms. The convergence result is applied to several examples as estimation...
On the convergence of moments in the CLT for triangular arrays with an application to random polynomials
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.
On the convergence of sequences of iterates of random-valued functions.
On the convergence of sequences of iterates of random-valued vector functions
Given a probability space (Ω,,P) and a subset X of a normed space we consider functions f:X × Ω → X and investigate the speed of convergence of the sequence (fⁿ(x,·)) of the iterates defined by f¹(x,ω ) = f(x,ω₁), .
On the convergence of stochastic integrals driven by processes converging on account of a homogenization property.
On the Convergence of the Critical Values of Uniformly Most Powerful Unbiased Tests for Two-Sided Hypotheses.
On the convergence of the ensemble Kalman filter
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.
On the difference between the product and the convolution product of distribution functions.
On the discretization in time of parabolic stochastic partial differential equations
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...
On the discretization in time of parabolic stochastic partial differential equations
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...
On the distribution density of the supremum of a random walk in the subexponential case.
On the distribution of random walk local time
On the distribution of the number of vertices in layers of random trees.
On the distributions of norms of weighted quantile processes
On the extremal behavior of sub-sampled solutions of stochastic difference equations.
On the Feller strong law of large numbers for fields of -valued random variables.
On the functional central limit theorem for stationary processes
On the increments of the principal value of Brownian local time.