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Register a new userA high dimensional Central Limit Theorem for martingales, with applications to context tree models
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We establish a central limit theorem for (a sequence of) multivariate martingales which dimension potentially grows with the length $n$ of the martingale. A consequence of the results are Gaussian couplings and a multiplier bootstrap for the maximum of a multivariate martingale whose dimensionality $d$ can be as large as $e^{n^c}$ for some $c>0$. We also develop new anti-concentration bounds for the maximum component of a high-dimensional Gaussian vector, which we believe is of independent interest. The results are applicable to a variety of settings. We fully develop its use to the estimation of context tree models (or variable length Markov chains) for discrete stationary time series. Specifically, we provide a bootstrap-based rule to tune several regularization parameters in a theoretically valid Lepski-type method. Such bootstrap-based approach accounts for the correlation structure and leads to potentially smaller penalty choices, which in turn improve the estimation of the transition probabilities.
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In the last decade, sequential Monte-Carlo methods (SMC) emerged as a key tool in computational statistics. These algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles. These particles and weights are generated recursively according to elementary transformations: mutation and selection. Examples of applications include the sequential Monte-Carlo techniques to solve optimal non-linear filtering problems in state-space models, molecular simulation, genetic optimization, etc. Despite many theoretical advances the asymptotic property of these approximations remains of course a question of central interest. In this paper, we analyze sequential Monte Carlo methods from an asymptotic perspective, that is, we establish law of large numbers and invariance principle as the number of particles gets large. We introduce the concepts of weighted sample consistency and asymptotic normality, and derive conditions under which the mutation and the selection procedure used in the sequential Monte-Carlo build-up preserve these properties. To illustrate our findings, we analyze SMC algorithms to approximate the filtering distribution in state-space models. We show how our techniques allow to relax restrictive technical conditions used in previously reported works and provide grounds to analyze more sophisticated sequential sampling strategies.
The Benjamini-Hochberg (BH) procedure remains widely popular despite having limited theoretical guarantees in the commonly encountered scenario of correlated test statistics. Of particular concern is the possibility that the method could exhibit bursty behavior, meaning that it might typically yield no false discoveries while occasionally yielding both a large number of false discoveries and a false discovery proportion (FDP) that far exceeds its own well controlled mean. In this paper, we investigate which test statistic correlation structures lead to bursty behavior and which ones lead to well controlled FDPs. To this end, we develop a central limit theorem for the FDP in a multiple testing setup where the test statistic correlations can be either short-range or long-range as well as either weak or strong. The theorem and our simulations from a data-driven factor model suggest that the BH procedure exhibits severe burstiness when the test statistics have many strong, long-range correlations, but does not otherwise.
This note gives a central limit theorem for the length of the longest subsequence of a random permutation which follows some repeating pattern. This includes the case of any fixed pattern of ups and downs which has at least one of each, such as the alternating case considered by Stanley in [2] and Widom in [3]. In every case considered the convergence in the limit of long permutations is to normal with mean and variance linear in the length of the permutations.
We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure cental limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.
In this paper, we study a high-dimensional random matrix model from nonparametric statistics called the Kendall rank correlation matrix, which is a natural multivariate extension of the Kendall rank correlation coefficient. We establish the Tracy-Widom law for its largest eigenvalue. It is the first Tracy-Widom law for a nonparametric random matrix model, and also the first Tracy-Widom law for a high-dimensional U-statistic.
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