User fkh - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:59:15Z http://mathoverflow.net/feeds/user/31275 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/122421#122421 Comment by fkh fkh 2013-02-21T19:01:04Z 2013-02-21T19:01:04Z Thanks. This seems to be beneficial. http://mathoverflow.net/questions/121238/uniform-law-of-large-numbers-for-martingale-difference Comment by fkh fkh 2013-02-10T03:38:30Z 2013-02-10T03:38:30Z For all $\theta \in \Theta$, $\xi_{tn} (\theta)$ is assumed martingale difference.