Consider a random variable $X\sim B(n,\frac 12)$. I'm trying to estimate the asymptotic behaviour of its central moments $E((X-\frac n2)^r)$, where $r$ is even and in the range $\Omega(1)\leq r\leq O(n)$, and $n$ goes to $\infty$.


I've looked at inequalities for central moments of sums of independent variables, but they seem too general. I'd be very grateful if someone can point me in the right direction.



High order central moments of a symmetric binomial variable
Let $M:=(E(X-n/2)^r)^{1/r}$. 
By Corollary 2 in **[[Latala](http://projecteuclid.org/euclid.aop/1024404522)]**, 
$M\sim S$, where $A\sim B$ means that $\frac1C\,B\le A\le C B$ for some universal positive constant $C$ and 
\begin{equation}
 S:=r\sup\{t(2n/r)^t\colon1/r\le t\le t\_\*\}, 
\end{equation}
where $t\_\*:=\frac12\wedge\frac nr$. It is not hard to see that $t(2n/r)^t$ increases in $t\in[1/r,t\_\*]$, and so, 
$S=rt\_\*(2n/r)^{t\_\*}$. 
Thus, 
$M\sim\sqrt{nr}$ if $r\le2n$ and $M\sim n(n/r)^{n/r}\sim n$ if $r\ge2n$. 
That is, $M\sim\sqrt{n(r\wedge n)}$. 


