Uniform and nonuniform estimates in the CLT for Banach valued dependent random variables
TL;DR: In this paper, a uniform estimate of the rate of convergence in the central limit theorem in certain Banach spaces for dependent random variables is established when the Gaussian measure of the ϵ-neighbourhood of the boundary of a set is proportional to ϵ and the third order moment is finite in the strong sense.
Abstract: A uniform estimate of the rate of convergence in the central limit theorem (CLT) in certain Banach spaces for dependent random variables is established when the Gaussian measure of the ϵ-neighbourhood of the boundary of a set is proportional to ϵ and the third order moment is finite in the strong sense. A uniform estimate in the CLT for Banach valued dependent random variables is carried out when the B -space is well behaved for a martingale transform.
Citations
More filters
TL;DR: In this article, a central limit theorem for a triangular array of row-wise independent Hilbert-valued random elements with finite second moment is proved under mild convergence requirements on the covariances of the row sums and the Lindeberg condition along the evaluations at an orthonormal basis.
Abstract: A Central Limit Theorem for a triangular array of row-wise independent Hilbert-valued random elements with finite second moment is proved under mild convergence requirements on the covariances of the row sums and the Lindeberg condition along the evaluations at an orthonormal basis. A Central Limit Theorem for real-valued martingale difference arrays is obtained under the conditional Lindeberg condition when the row sums of conditional variances converge to a (possibly degenerate) constant. This result is then extended, first to multi-dimensions and next to Hilbert-valued elements, under appropriate convergence requirements on the conditional and unconditional covariances and the conditional Lindeberg condition along (orthonormal) basis evaluations. Extension to include Banach- (with a Schauder basis) valued random elements is indicated.
58 citations
Posted Content•
TL;DR: In this article, the authors focus on a single large Bayesian game and develop a bootstrap inference method that relaxes the assumption of rational expectations and allows for the players to form beliefs differently from each other.
Abstract: Econometric models of strategic interactions among people or firms have received a great deal of attention in the literature. Less attention has been paid to the role of the underlying assumptions about the way agents form beliefs about other agents. This paper focuses on a single large Bayesian game and develops a bootstrap inference method that relaxes the assumption of rational expectations and allows for the players to form beliefs differently from each other. By drawing on the main intuition of Kalai(2004), we introduce the notion of a hindsight regret, which measures each player's ex post value of other players' type information, and obtain its belief-free bound. Using this bound, we derive testable implications and develop a bootstrap inference procedure for the structural parameters. We demonstrate the finite sample performance of the method through Monte Carlo simulations.
7 citations
Cites background from "Uniform and nonuniform estimates in..."
...Since λi,ρ(Ti) is small and u ∆ i (1) depends on Y−i only through the within group proportion 1 Ns−1 ∑ i∈Ns\{i} Yi, this difference becomes negligible by Assumptions 1 and 2....
[...]
...3) 1 {Yi = 1} ≤ 1 { Eyi [ ui (1)|Ti ] ≥ 0 } ....
[...]
...Then ei,U is close to the difference between P{Eyi [ui (1)|Ti] ≥ 0|Xi} and P{ui (1) ≥ −λi,ρ(Ti)|Y−i, Xi}....
[...]
...Then ei,U is close to the difference between P{Eyi [u∆i (1)|Ti] ≥ 0|Xi} and P{u∆i (1) ≥ −λi,ρ(Ti)|Y−i, Xi}....
[...]
...5) 1− πi,L ≤ P {Yi = 1|Xi} ≤ πi,U , where πi,U ≡ P { Eyi [ ui (1)|Ti ] ≥ 0|Xi } and πi,L ≡ P { Eyi [ ui (1)|Ti ] ≤ 0|Xi } ....
[...]
Posted Content•
TL;DR: In this paper, the authors considered a stationary, linear Hilbert space valued process and established Berry-Essen type results with optimal convergence rates under sharp dependence conditions on the underlying coefficient sequence of the linear operators.
Abstract: Consider a stationary, linear Hilbert space valued process. We establish Berry-Essen type results with optimal convergence rates under sharp dependence conditions on the underlying coefficient sequence of the linear operators. The case of non-linear Bernoulli-shift sequences is also considered. If the sequence is $m$-dependent, the optimal rate $(n/m)^{1/2}$ is reached. If the sequence is weakly geometrically dependent, the rate $(n/\log n)^{1/2}$ is obtained.
2 citations
Cites background from "Uniform and nonuniform estimates in..."
...Certain martingale difference sequences in Banach spaces have been investigated in [1], [10]....
[...]
1 citations
Posted Content•
TL;DR: In this article, conditional Berry-Esseen bounds for the maximum of the sum of a vector-valued martingale difference sequence adapted to a filtration were provided for Gaussian random vectors.
Abstract: This note provides conditional Berry-Esseen bounds for the maximum of the sum of a vector-valued martingale difference sequence $\{X_i\}_{i=1}^n$ adapted to a filtration $\{\mathcal{F}_i\}_{i=1}^n$. We approximate the conditional distribution of the maximum of $S=\sum_{i\le n}X_i$ given some $\sigma$-field $\mathcal{F}_0\subset \mathcal{F}_1$ by that of the maximum of a mean-zero normal random vector having the same conditional variance given $\mathcal{F}_0$ as the vector $S$. In order to overcome the non-differentiability of the maximum function, we approximate the latter by a version of the log sum of exponentials. This work relies on the anti-concentration bound for maxima of Gaussian random vectors established in Chernozhukov et al. (2015).
References
More filters
Book•
01 Jan 1967
TL;DR: The Borel subsets of a metric space Probability measures in the metric space and probability measures in a metric group Probability measure in locally compact abelian groups The Kolmogorov consistency theorem and conditional probability probabilistic probability measures on $C[0, 1]$ and $D[0-1]$ Bibliographical notes Bibliography List of symbols Author index Subject index as mentioned in this paper
Abstract: The Borel subsets of a metric space Probability measures in a metric space Probability measures in a metric group Probability measures in locally compact abelian groups The Kolmogorov consistency theorem and conditional probability Probability measures in a Hilbert space Probability measures on $C[0,1]$ and $D[0,1]$ Bibliographical notes Bibliography List of symbols Author index Subject index.
2,667 citations
TL;DR: In this article, it was shown that Chung's version of the strong law of large numbers holds, if and only if $E$ is of type p. If the variables are identically distributed, then the central limit theorem is valid.
Abstract: Let $X_1, X_2, \cdots$ be independent random variables with values in a Banach space $E$. It is then shown that Chung's version of the strong law of large numbers holds, if and only if $E$ is of type $p$. If the $X_n$'s are identically distributed, then it is shown that the central limit theorem is valid, if and only if $E$ is of type 2. Similar results are obtained for vectorvalued martingales.
283 citations
01 Jan 1972
TL;DR: In this article, it was shown that the necessary and sufficient conditions for convergence in distribution to any specified, infinitely divisible law remain sufficient also in the most general dependent case, provided that quantities such as means and the like, are replaced by conditional means, and the conditioning being relative to the preceding sum.
Abstract: 1.1. Limiting distributions of sums of independent random variables have been exhaustively studied and there is a satisfactory general theory of the subject (see the monograph of B. Gnedenko and A. Kolmogorov [6], or advanced text books on probability theory such as that of M. Loeve [10]). Our knowledge of the corresponding theory for dependent random variables is much more meagre. Although a great number of papers have been published on the subject, not many general results are known. In recent years the author has shown ([4], [5]) that the necessary and sufficient conditions for convergence in distribution to any specified, infinitely divisible law remain sufficient also in the most general dependent case, provided that quantities such as means and the like, are replaced by conditional means, and the like, the conditioning being relative to the preceding sum. (The necessity of the conditions requires, in general, further assumptions.) In the present paper we are concerned almost exclusively with asymptotic normality. Though our general results about asymptotic normality can be obtained by direct specialization of the results mentioned above, we preferred to develop them here independently. We hope that the greater accessibility of the present proofs will compensate for this sacrifice of brevity. After establishing the general results we give a few applications. It would be quite easy to extend the list of applications indefinitely by going through various results in the literature and seeing how they can be improved by using our general theorems. 1.2. We consider random variables arranged in a double array
159 citations
42 citations
TL;DR: In this article, a generalization of Edgeworth expansion for functions of normalized sums of i.i.d. vectors is defined, which is valid up to √ n √ s − 2/2 for functions with bounded Frechet derivatives.
Abstract: In this paper we define a generalization of Edgeworth expansions for the expectation of functions of normalized sums of i.i.d. Banach space valued random vectors. These expansions are valid up to $0(n^{-(s - 2)/2})$ for functions with $3(s - 2)$ bounded Frechet derivatives and random vectors with finite $s^{th}$ absolute moment.
30 citations