User dennis - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-27T02:55:49Z http://mathoverflow.net/feeds/user/30524 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118476/prove-an-inequality-related-to-moments Prove an inequality related to moments Dennis 2013-01-09T20:28:38Z 2013-01-12T22:58:14Z <p>I am reading a paper and stuck with an inequality used in that paper. </p> <p>$\varepsilon^n=(\varepsilon_1^n, \varepsilon_2^n,\ldots,\varepsilon_n^n)^T$ is a vector of i.i.d. random variables with mean 0 and variance $\sigma^2$. Assume that $\varepsilon_i^n$ have finite $2k$'th moment $E(\varepsilon_i^n)^{2k}&lt;\infty$ for an integer $k>0$. Show that for a constant $n$-dimensional vector $\alpha$, have</p> <p>$$E(\alpha^T\varepsilon^n)^{2k}\leq (2k-1)!!\|\alpha\|_2^2E(\varepsilon_i^n)^{2k}.$$</p> <p>The paper I am reading is "On Model Selection Consistency of Lasso" by Zhao and Yu 2006, which can be found via <a href="http://jmlr.csail.mit.edu/papers/volume7/zhao06a/zhao06a.pdf" rel="nofollow">http://jmlr.csail.mit.edu/papers/volume7/zhao06a/zhao06a.pdf</a> and the inequality appears on Page 2558.</p> <p>Thanks</p> http://mathoverflow.net/questions/118476/prove-an-inequality-related-to-moments/118769#118769 Comment by Dennis Dennis 2013-01-14T02:42:25Z 2013-01-14T02:42:25Z Thank you! It's clear and helpful.