Uniform law of large numbers for martingale difference - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:06:43Z http://mathoverflow.net/feeds/question/121238 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121238/uniform-law-of-large-numbers-for-martingale-difference Uniform law of large numbers for martingale difference fkh 2013-02-08T20:58:40Z 2013-02-20T16:53:14Z <p>Let $\xi_{tn}(\theta),t=1,\dots,n$ be a real-valued martingale difference array indexed by a parameter $\theta \in \Theta \subset R$, where the set $\Theta$ is compact. Now, for all fixed $\theta \in \Theta$, the law of large numbers, </p> <p>$\sum_{t=1}^n \xi_{tn} (\theta) \rightarrow 0$ in probability, </p> <p>is assumed to hold. Then, what conditions make the following UNIFORM law of large numbers hold?</p> <p>$\sup_{\theta \in \Theta} | \sum_{t=1}^n \xi_{tn}(\theta) | \rightarrow 0$ in probability. </p> <p>Are there any articles or books about the result? </p> http://mathoverflow.net/questions/121238/uniform-law-of-large-numbers-for-martingale-difference/121535#121535 Answer by Yuri Bakhtin for Uniform law of large numbers for martingale difference Yuri Bakhtin 2013-02-11T21:57:08Z 2013-02-11T21:57:08Z <p>Even if $\xi$'s are independent (a specific case of your martingale-difference stitation), uniform LLN sometimes holds and sometimes it does not. This type of questions has been studied in Machine learning and, specifically, in Vapnik-Chervonenkis theory. The Glivenko-Cantelli theorem (see wikipedia) describes one situation where the convergence is uniform. More general results can be formulated in terms of VC (Vapnik-Chervonenkis) classes. Perhaps some literature is available for martingales, too.</p> http://mathoverflow.net/questions/121238/uniform-law-of-large-numbers-for-martingale-difference/122421#122421 Answer by alr for Uniform law of large numbers for martingale difference alr 2013-02-20T16:53:14Z 2013-02-20T16:53:14Z <p>An extension of the uniform Glivenko-Cantelli notion to martingale differences can be found here: www-stat.wharton.upenn.edu/~rakhlin/papers/emp_proc_dep.pdf</p>