Krdytkyu

If $X$ is a real-valued random variable, then the $k$th raw moment is $mathbb{E}[X^k]$, while the $k$th central moment is $mathbb{E}[(X-mathbb{E}[X])^k]$. If $k$ is even, is the $k$th central moment always bounded above the $k$th raw moment?

When $k = 2$, then $mathbb{E}[(X-mathbb{E}[X])^2] = mathbb{E}[X^2]-mathbb{E}[X]^2$, and because $mathbb{E}[X]^2$ is always positive, it follows that this is less than or equal to $mathbb{E}[X^2]$. But I'm having trouble extending this to larger moments.

The statement is not true in general and it is easy to construct a counterexample. For example, we want $mathbb{E}(X^4)leq mathbb{E}((X-mathbb{E}(X))^4)$, we can let $X$ has a small probability of taking a large positive value, while keeping $mathbb{E}(X)$ negative. Let
$X$ be a random variable with $P(X=-2)=1-epsilon$ and $P(x=M)=epsilon$. We choose $epsilon=1/(M+2)$ such that $mathbb{E}(X)=-1$.

Then we have
$$
mathbb{E}(X^2)=M+2,quad mathbb{E}((X-mathbb{E}(X))^2)=M+1
$$
and
$$
mathbb{E}(X^4)=M^3-2M^2+4M+8,quad mathbb{E}((X-mathbb{E}(X))^4)=M^3+2M^2+2M+1.
$$
Obviously, if $M$ is large enough, we have $mathbb{E}(X^4)leq mathbb{E}((X-mathbb{E}(X))^4)$.

To complement the accepted answer,
we can show that $E[(X - E[X])^m] leq E[X^m]$ when $X geq 0$.
Assume without loss of generality that $E[X] = 1$ (rescaling $X$ if necessary).
Then, omitting the dependence on $omega$ for notational convenience,
begin{align}
& int_{Omega} X^m , d Omega - int_{Omega} (X - 1)^m , dP(omega) \
& quad =
int_{X leq 1/2} left[ X^m - (X - 1)^m right] , dP(omega) +
int_{1/2 < X leq 1} left[ X^m - (X - 1)^m right] , dP(omega) \
& quad + int_{X > 1} left[ X^m - (X - 1)^m right] , dP(omega). \
& quad geq
int_{X leq 1/2} left[ X^m - (X - 1)^m right] , dP(omega)
+ int_{X > 1} left[ X^m - (X - 1)^m right] , dP(omega)
=: I_1 + I_2.
end{align}
For $m$ odd, both $I_1$ and $I_2$ are positive and the statement is proved.
We assume from now on that $m$ is even.
It is easy to see that:
begin{align}
|a^m - (a - 1)^m| leq (1 - a) quad & text{ if } quad 0 leq a leq 1/2; \
a^m - (a - 1)^m geq (a - 1) quad & text{ if } quad a > 1.
end{align}
Using these inequalities, we deduce that
$$ I_1 + I_2 geq int_{X > 1} (X - 1) , dP(omega) - int_{X leq 1/2} (1 - X) , dP(omega). $$
To conclude, we note that,
by definition of the expected value,
begin{align}
& int_{X leq 1/2} (X(omega) - 1) , dP(omega) + int_{1/2 < X leq 1} (X(omega) - 1) , dP(omega) + int_{X > 1} (X(omega) - 1) , dP(omega) = 0 \
& rightarrow int_{X > 1} (X(omega) - 1) , dP(omega) geq int_{X leq 1/2} (1 - X(omega)) , dP(omega).
end{align}