Applying moment inequalities to random variables

Question

So I am looking at this paper that applies some moment inequalities like the Chebyshev's inequality to an expression like the one on the left hand side and yields the right hand side below $$\dfrac{\#\{i: 1\leq i\leq n, |X_i|>c\}}{n}\leq \dfrac{1/n \sum_i X_i^2}{c^2}.$$ The problem is the Chebyshev's inequality that I know of bounds probabilities and we have a random variable in the numerator. Since there is a $n$ in the bottom I do see how it may become a probability but I am not sure how to apply the inequality directly.

Answer 1

I guess this is one way to look at it:

By this version of Markov's inequality we have $\Pr\left(|X_i|>c\right)\le\dfrac{\mathrm{E}(|X_i|^2)}{c^2}=\dfrac{\mathrm{E}(X_i^2)}{c^2}$, $1\le i\le n$.

The probability in the l.h.s is estimated by the proportion of $x_i$'s satisfying $|x_i|>c$ in a sample of size $n$. This proportion is precisely $\displaystyle\frac{\#\{i: 1\leq i\leq n, |x_i|>c\}}{n}$, where the $\#$ indicates count. Similarly, the expected value $\mathrm{E}(X_i^2)$ is estimated by $\displaystyle\frac{1}{n}\sum_{i=1}^n x_i^2$ from the random sample.