Moments of a distribution - any use for partial or higher moments? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:24:40Z http://mathoverflow.net/feeds/question/39387 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39387/moments-of-a-distribution-any-use-for-partial-or-higher-moments Moments of a distribution - any use for partial or higher moments? Edward 2010-09-20T14:17:04Z 2010-09-20T19:39:08Z <p>It is usual to use second, third and fourth moments of a distribution to describe certain properties. Do partial moments or moments higher than fourth describe any useful properites of a distribtution.</p> http://mathoverflow.net/questions/39387/moments-of-a-distribution-any-use-for-partial-or-higher-moments/39405#39405 Answer by Sivan Sabato for Moments of a distribution - any use for partial or higher moments? Sivan Sabato 2010-09-20T17:09:47Z 2010-09-20T17:09:47Z <p>That depends on what you mean by "useful properties". For instance, all of the even moments are required to characterize sub-Gaussian random variables, which are those variables whose tail is majorized by that of a Gaussian random variable. A random variable X is sub-Gaussian if and only if there exists a non-negative number b such that $E[X^{2k}] \leq \frac{(2k)!b^{2k}}{2^k k!}$ for all $k \geq 1$.</p> http://mathoverflow.net/questions/39387/moments-of-a-distribution-any-use-for-partial-or-higher-moments/39416#39416 Answer by Spencer for Moments of a distribution - any use for partial or higher moments? Spencer 2010-09-20T18:43:24Z 2010-09-20T18:43:24Z <p>The typical example I remember being given at school is that of <a href="http://en.wikipedia.org/wiki/Kurtosis" rel="nofollow">kurtosis</a>, which uses fourth moment and is easy to understand. </p> <p>Wiki says: "Higher kurtosis means more of the variance is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations."</p> http://mathoverflow.net/questions/39387/moments-of-a-distribution-any-use-for-partial-or-higher-moments/39422#39422 Answer by Yemon Choi for Moments of a distribution - any use for partial or higher moments? Yemon Choi 2010-09-20T19:39:08Z 2010-09-20T19:39:08Z <p>Not sure if this is what you had in mind, but a bounded random variable (i.e. one which almost surely takes values in some bounded subset of $\mathbb R$) is uniquely determined by its moments.</p> <p>Off the top of my head, one way to see this - probably not the most efficient! - is that the distribution of a random variable $X$ is <a href="http://en.wikipedia.org/wiki/Characteristic_function_%2528probability_theory%2529" rel="nofollow">uniquely determined</a> by the set of values ${\mathbb E}f(X)$ as $f$ runs over all bounded continuous functions $\mathbb R\to \mathbb R$; and now since $X$ is a.s. bounded, the <a href="http://en.wikipedia.org/wiki/Stone%25E2%2580%2593Weierstrass_theorem#Weierstrass_approximation_theorem" rel="nofollow">Weierstrass approximation theorem</a> implies that $X$ is uniquely determined by the set of values $\mathbb E p(X)$ as $p$ runs over all polynomials.</p> <p>Thus in some settings, knowing the moments is equivalent to knowing the distribution. (This point of view is much loved by those working in "free probability", but that's another story altogether...)</p> <p>I should also mention that the question of just when we can say that the moment sequence determines the distribution has also been studied: see <a href="http://en.wikipedia.org/wiki/Carleman%2527s_condition" rel="nofollow">here</a> for instance.</p>