Approximation Thorems for Independent and Weakly Dependent Random Vectors
István Berkes,Walter Philipp +1 more
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In this paper, approximation theorems of the following type were proved for sums of independent identically distributed random variables with values in Θ( √ √ n) with a logarithmic mixing rate.Abstract:
In this paper we prove approximation theorems of the following type. Let $\{X_k, k \geqslant 1\}$ be a sequence of random variables with values in $\mathbb{R}^{d_k}, d_k \geqslant 1$ and let $\{G_k, k \geqslant 1\}$ be a sequence of probability distributions on $\mathbb{R}^{d_k}$ with characteristic functions $g_k$ respectively. If for each $k \geqslant 1$ the conditional characteristic function of $X_k$ given $X_1, \cdots, X_{k - 1}$ is close to $g_k$ and if $G_k$ has small tails, then there exists a sequence of independent random variables $Y_k$ with distribution $G_k$ such that $|X_k - Y_k|$ is small with large probability. As an application we prove almost sure invariance principles for sums of independent identically distributed random variables with values in $\mathbb{R}^d$ and for sums of $\phi$-mixing random variables with a logarithmic mixing rate.read more
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Book ChapterDOI
Basic Properties of Strong Mixing Conditions
TL;DR: A survey of the properties of strong mixing conditions for sequences of random variables can be found in this paper, where the focus will be on the structural properties of these conditions, and not at all on limit theory.
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Empirical Process Techniques for Dependent Data
Herold Dehling,Walter Philipp +1 more
TL;DR: In this article, the authors provide a survey of classical and modern techniques in the study of empirical processes of dependent data, and provide necessary technical tools like correlation and moment inequalities, and prove central limit theorems for partial sums.
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On U-statistics and v. mise’ statistics for weakly dependent processes
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TL;DR: In this article, the authors extend these results considerably and prove central limit theorems and their rate of convergence (in the Prohorov metric and a Berry Esseen type theorem), functional central limit theorem and as approximation by a Brownian motion.
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TL;DR: In this paper, a multivariate framework for terminating simulation in MCMC is presented, which requires strongly consistent estimators of the covariance matrix in the Markov chain central limit theorem (CLT), and a lower bound on the number of minimum effective samples required for a desired level of precision.
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The Wasserstein distance and approximation theorems
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