No Arabic abstract
Any (measurable) function $K$ from $mathbb{R}^n$ to $mathbb{R}$ defines an operator $mathbf{K}$ acting on random variables $X$ by $mathbf{K}(X)=K(X_1, ldots, X_n)$, where the $X_j$ are independent copies of $X$. The main result of this paper concerns selectors $H$, continuous functions defined in $mathbb{R}^n$ and such that $H(x_1, x_2, ldots, x_n) in {x_1,x_2, ldots, x_n}$. For each such selector $H$ (except for projections onto a single coordinate) there is a unique point $omega_H$ in the interval $(0,1)$ so that for any random variable $X$ the iterates $mathbf{H}^{(N)}$ acting on $X$ converge in distribution as $N to infty$ to the $omega_H$-quantile of $X$.
In this paper, we prove convergence in distribution of Langevin processes in the overdamped asymptotics. The proof relies on the classical perturbed test function (or corrector) method, which is used both to show tightness in path space, and to identify the extracted limit with a martingale problem. The result holds assuming the continuity of the gradient of the potential energy, and a mild control of the initial kinetic energy.
We consider the probability distributions of values in the complex plane attained by Fourier sums of the form sum_{j=1}^n a_j exp(-2pi i j nu) /sqrt{n} when the frequency nu is drawn uniformly at random from an interval of length 1. If the coefficients a_j are i.i.d. drawn with finite third moment, the distance of these distributions to an isotropic two-dimensional Gaussian on C converges in probability to zero for any pseudometric on the set of distributions for which the distance between empirical distributions and the underlying distribution converges to zero in probability.
We study the weak limits of solutions to SDEs [dX_n(t)=a_nbigl(X_n(t)bigr),dt+dW(t),] where the sequence ${a_n}$ converges in some sense to $(c_- 1mkern-4.5mumathrm{l}_{x<0}+c_+ 1mkern-4.5mumathrm{l}_{x>0})/x+gammadelta_0$. Here $delta_0$ is the Dirac delta function concentrated at zero. A limit of ${X_n}$ may be a Bessel process, a skew Bessel process, or a mixture of Bessel processes.
The Robbins-Monro algorithm is a recursive, simulation-based stochastic procedure to approximate the zeros of a function that can be written as an expectation. It is known that under some technical assumptions, a Gaussian convergence can be established for the procedure. Here, we are interested in the local limit theorem, that is, quantifying this convergence on the density of the involved objects. The analysis relies on a parametrix technique for Markov chains converging to diffusions, where the drift is unbounded.
We address the problem of proving a Central Limit Theorem for the empirical optimal transport cost, $sqrt{n}{mathcal{T}_c(P_n,Q)-mathcal{W}_c(P,Q)}$, in the semi discrete case, i.e when the distribution $P$ is finitely supported. We show that the asymptotic distribution is the supremun of a centered Gaussian process which is Gaussian under some additional conditions on the probability $Q$ and on the cost. Such results imply the central limit theorem for the $p$-Wassertein distance, for $pgeq 1$. Finally, the semidiscrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials.