# Control high order moments by lower order moment?

Let $X$ be a random variable and suppose that we have $E(|X|) <+\infty$ and $E(|X|^2)<+\infty$. In particulary, I can find a constant C such that $$E(|X|^2) < C\quad\quad (1)$$ I consider the higher moments of $X$ as $E(|X|^{2p})$ where $p>1$. I have two questions:

1) Can we always conclude that $E(|X|^{2p}) < + \infty$ by using (1) (maybe not!) or it depends on the properties of $X$?

2) Suppose that $E(|X|^{2p}) < + \infty$ and if I want to find an upper bound for $E(|X|^{2p})$. Is there any inequality that allow us to find the upper bound by using (1)?

• Answer is no to both questions when p>1. But the answer is yes when p <1 – Amr Mar 24 '16 at 11:42
• For example, knowing $E(|X|^4)$ puts a natural upper bound on $E(|X|^2)$ (the square root) but not the other way round. – Henry Mar 24 '16 at 15:38

## 1. We can have $$\mathbb E(X^{2p})=\infty$$
Let us assume $$P(X)\propto\frac1{|X|^\nu+1}$$. If $$\nu>1$$, it is a well defined probability distribution, which has a finite $$\mathbb{E}(|X|)$$ for $$\nu>2$$ and a finite $$\mathbb{E}(|X|^2)$$ for $$\nu>3$$.
Actually we have \begin{align} \mathbb{E}(|X|^\mu)&\text{ finite} & \text{if }&\mu<\nu\\ \mathbb{E}(|X|^\mu)&=+\infty & \text{if }&\mu\ge\nu, \end{align} therefore, a good choice of $$\nu$$ allows set the rank after which all moments are infinite, even if all the lower moments are bounded
## 2. We can have really high finite $$\mathbb E(X^{2p})$$, even when $$\mathbb E(X^{2p})\le C$$
Just look at the following simple probability distribution for $$X$$, defined for $$\varepsilon>0$$: \begin{align} P(X=0)&=1-\varepsilon; & P\left(X=\sqrt{\frac C\varepsilon}\right)&=\varepsilon. \end{align} We have then, assuming $$\varepsilon\ll 1$$ and $$p>1$$ \begin{align} \mathbb E(|X|)&=\sqrt{\varepsilon C}&\ll \sqrt{C}\\ \mathbb E(|X|^2)&=\varepsilon C&\\ \mathbb E(|X|^{2p})&=\frac{C^p}{\varepsilon^{p-1}}& \gg C^p \end{align} Chosing a small enough $$\varepsilon$$ allows to choose a $$E(|X|^{2p})$$ arbitrary large, but finite, while keeping the first two moments below $$C$$.