User gmravi2003 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T13:53:15Z http://mathoverflow.net/feeds/user/12586 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120794/maximal-inequalities-for-certain-functions-of-a-martingale-difference-sequence Maximal inequalities for certain functions of a martingale difference sequence gmravi2003 2013-02-04T19:11:37Z 2013-02-06T02:42:26Z <p>Suppose $\xi_1,\ldots \xi_T$ is a martingale difference sequence. Then,</p> <p>1) For any $a\in \mathbb{R}^{+}$, can we say something about the sequence $\xi_1^2\mathbb{1}(\xi_1\geq a),\ldots, \xi_T^2\mathbb{1}(\xi_T\geq a)$ ? Is it a (sub/sup) martingale difference sequence?</p> <p>2) Suppose each $|\xi_t|\leq B_t$ a.s., and $B_1\leq B, B_T\leq B$ a.s. then can we provide an upper bound for $\mathbb{P}(\sum_{t=1}^T \xi_t\geq z)$?</p> <p>I guess if we can prove that the sequence in (1) is a (sub/sup) martingale difference sequence then one can apply standard maximal inequalities to solve (2). However I am not able to resolve (1), and my intuition says that, for the sequence in (1) one cannot claim any (sub/sup) martingale difference behaviour. However I do not have a formal proof or a counterexample. Also if it turns out that the sequence in (2) is not a (sub/sup) martingale then how do we go about establishing maximal inequalities?</p> http://mathoverflow.net/questions/96222/intersection-of-an-uncountable-number-of-sets Intersection of an uncountable number of sets. gmravi2003 2012-05-07T14:39:16Z 2012-12-08T22:49:26Z <p>Let $\mathcal{I}$ be an uncountable set. Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, and $E_i, i\in \mathcal{I}$ be a measurable set such that $\mathbb{P}(E_i)=1$. What can we say about $\mathbb{P}(\cap_{i\in \mathcal{I}} E_i)?$. </p> <p>I know that an uncountable intersection of measurable sets is not necessarily measurable. But are there in general conditions that allow such infinite intersections to be measurable? </p> <p>Apologies if the question is too elementary or unclear. </p> http://mathoverflow.net/questions/96711/does-the-minima-of-a-sequence-of-convex-convergent-functions-converge Does the minima of a sequence of convex convergent functions converge? gmravi2003 2012-05-11T21:06:43Z 2012-05-12T04:46:12Z <p>Suppose $f_1,f_2,\ldots$ is a sequence of convex functions that converges to a continuous convex $f$. Let $x_1^*,x_2^*$ be their respective (not necessarily unique) minima, and let y be a minima of $f$ (once again need not be unique). Can we prove that there exists a version of $x_1^*,x_2^*,\ldots$ such that $x_n^*\rightarrow y$ ? </p> http://mathoverflow.net/questions/120794/maximal-inequalities-for-certain-functions-of-a-martingale-difference-sequence/120821#120821 Comment by gmravi2003 gmravi2003 2013-02-06T02:46:38Z 2013-02-06T02:46:38Z Thanks Alex for your reply. Sorry there was a typo in question (2). It should have read as $|\xi_t| \leq B_t$ a.s., and $B1≤B,B_2\leq B, \ldots B_T\leq B$. What is not clear to me though is 1) Why is the first sequence a submartingale difference sequence. 2) Azuma-Hoeffding (AH) bound applies only to (sup)martingales. If the first sequence is a submartingale, then I do not see how one could apply AH? http://mathoverflow.net/questions/96711/does-the-minima-of-a-sequence-of-convex-convergent-functions-converge/96717#96717 Comment by gmravi2003 gmravi2003 2012-05-11T22:06:34Z 2012-05-11T22:06:34Z Thanks Will. The last paragraph is unclear to me. To start did you mean if f has a unique minima? http://mathoverflow.net/questions/96222/intersection-of-an-uncountable-number-of-sets/96233#96233 Comment by gmravi2003 gmravi2003 2012-05-09T02:20:04Z 2012-05-09T02:20:04Z I guess your theorem allows us to conclude that $\cap_{i \in \mathcal{I}} E_i=E$?, but doesn't provide us with the measure of set E?