Khinchin's weak law of large numbers: finite variance

Question

I have the following situation: suppose you have a sequence of i.i.d. random variables $\{X_i\}$ with mean $\mu$ and variance $1$.

I would like to use Khinchin's WLLN on it, but this requires that $\mathbb{E}(|X_i|)<\infty$. I clearly see that the finite variance implies $\mathbb{E}(X_i^2)=\mu^2+1<\infty$, but I cannot convince myself that this implies the expected value of absolute value of my r.v. is also finite.

Any ideas?

Answer 1

Let $0<r<s$. Using Hölder's inequality, we obtain that $$ \operatorname E|X|^r\le\bigl(\operatorname E|X|^s\bigr)^{r/s}. $$

So if a random variable $X$ has a finite absolute moment of order $s$, then it also has a finite absolute moment of any order less than $s$.