Showing papers by "Miguel A. Arcones published in 1994"
TL;DR: In this article, limit theorems for functions of stationary mean-zero Gaussian sequences of vectors satisfying long range dependence conditions are considered and a sufficient bracketing condition for these limit-theorems to happen uniformly over a class of functions is presented.
Abstract: Limit theorems for functions of stationary mean-zero Gaussian sequences of vectors satisfying long range dependence conditions are considered. Depending on the rate of decay of the coefficients, the limit law can be either Gaussian or the law of a multiple Ito-Wiener integral. We prove the bootstrap of these limit theorems in the case when the limit is normal. A sufficient bracketing condition for these limit theorems to happen uniformly over a class of functions is presented.
236 citations
TL;DR: In this article, a uniform central limit theorem for weak convergence to Gaussian processes of empirical processes and U-processes from stationary β mixing sequences indexed by V-C subgraph classes of functions is given.
Abstract: This paper gives sufficient conditions for the weak convergence to Gaussian processes of empirical processes andU-processes from stationary β mixing sequences indexed byV-C subgraph classes of functions. If the envelope function of theV-C subgraph class is inL
p
for some 21. These conditions are almost minimal.
168 citations
TL;DR: In this article, it was shown that Liu's simplical median and Oja's medians are asymptotically normal under suitable conditions, and this was then applied to prove asymPT normality of Liu's median and of Oja medians in the Euclidean space.
Abstract: If a criterion function $g(x_1, \ldots, x_m; \theta)$ depends on $m > 1$ samples, then a natural estimator of $\arg \max P^mg(x_1, \ldots, x_m; \theta)$ is the $\arg \max$ of a $U$-process. It is observed that, under suitable conditions, these estimators are asymptotically normal. This is then applied to prove asymptotic normality of Liu's simplical median and of Oja's medians in $\mathbb{R}^d$.
65 citations
TL;DR: In this paper, exponential inequalities, the law of the iterated logarithm and the bootstrap central limit theorem for U-processes indexed by Vapnik-Cervonenkis classes of functions are derived.
44 citations
TL;DR: In this article, the convergence rate of the Lp-medians for 0 1 2 was shown to be at most n 1 2, whereas the rate of convergence of M-estimators is n 1 3.
16 citations
TL;DR: In this paper, some laws of the iterated logarithm for empirical processes rescaled in the "time" parameter are presented and applied to obtain strong limit theorems for M-estimators.
13 citations
TL;DR: In this paper, it was shown that the limite law of canonical U-process is the law of a chaos process which has a versio with bounded and 2 continuous paths, and that this is also true for B-valued canonicalU-statistics with values in a separable Banach space.
Abstract: It is shown that the limite law of canonicalU-process is the law of a chaos process which has a versio with bounded and ‖·‖2 continuous paths. This is also true forB-valued canonicalU-statistics with values in a separable Banach space. Some properties of Banach spaces of type 2 related withU-statistics are presented.
10 citations
TL;DR: In this paper, it was shown that the condition ∑∞ k = 1 (k−1 1 2, which is known to be equivalent to F 1 being P-Donsker, implies that F α is P-donsker for 1 2.
8 citations
01 Jan 1994
TL;DR: In this article, an M-estimator θn is defined as a r.i.d. r.v. estimator with values in S. The objective is to estimate a parameter θ0 characterized by E[h(X,θ 0)] = 0.
Abstract: Let (S,S) be a measurable space, let Θ be a subset of ℝm and let h : S × Θ → ℝd be a jointly measurable funcion. Further, let X and {X i}i=1 ∞ be i.i.d. r.v.’s with values in S. Let P be the distribution of X. Suppose that we want to estimate a parameter θ0 characterized by E[h(X,θ 0)] = 0. An M-estimator θn is a r.v. θ n = θn X 1,…, X n) satisfying
$$n^-1\sum_{i=0}^{n} h(X_i,\theta_n)\approx 0$$
.
5 citations