Very large moments of binomial distribution

Question

Is there a bound on the very large moments of binomial distribution, in the asymptotic sense? In other words, is there a bound on $\left(\mathbb{E}\left[ X^k\right]\right)^{1/k}$ or an asymptotic behavior result for $\mathbb{E}\left[ X^k\right]$ as $k \to \infty$, where $X\sim Binomial(n,p)$ and $n$ and $p$ are fixed? I did some numerical experiments and it seems like $\left(\mathbb{E}\left[ X^k\right]\right)^{1/k} \to 2np$, but could not find a way to prove this.

Answer 1

Yes, Jensen's inequality says the $L^p$ norms $$ |X|_p = E(|X|^p)^{1/p}$$ are increasing in $p$. It turns out there's an $L^{\infty}$ norm $$ |X|_{\infty} = \inf\{K : P(|X| \le K) = 1\}$$ that "lives up to its name" in that $\lim_{p\rightarrow\infty}|X|_p = |X|_{\infty}$

So it converges to $n$, not $2pn$. Maybe you had $p=1/2$? Or you might be seeing dependence on $p$ due to undersampling the tail. Basically, the higher you take $k$ the more very rare events in the tail matter, so they are what push the expected value up to its maximum possible value of $n.$ But if you don't run your simulation long enough you won't see enough of these rare events for your moment to converge well statistically.

Your formula can't be right. For instance if $p=1$ ($p$ as in the probability of success, not $L^p$) you should have $$E(X^k)^{1/k}= (n^k)^{1/k} =n$$ which converges as $k\rightarrow\infty$ to $n,$ not $2pn=2n.$

(Note the theorem about $L^\infty$ is generic... no assumption of the binomial distribution here.)