Error Term of Chebyshev inequality?

Question

Chebyshev inequality tells us that $$Pr[|X-E[X]|\geq a]\leq \frac{Var[X]^2}{a^2}$$

Do you know an Expression (or a paper where this Expression is mentioned) for the error term?

Answer 1

Unfortunately Chebyshevs inequality is tight, in the sense that there are many random variables which turn the inequality into an equality. For practice purposes if you want a better bound, typically you need higher moments to exist and grow slowly. In other words:

$$P(|X-EX|>c)\leq\frac{E[(X-EX)^k]}{c^k}$$

On the other hand there is an exact statement about the error. It comes directly from the proof of the inequality. Let $Y:=(X-EX)^2$ Then

$$\mbox{Var}(X)=E[Y]=\int_{Y\geq c}YdP+\int_{Y<c}YdP\geq cP(Y\geq c)$$

Where it's clear what was thrown away: the under estimate of the first integral and the entire second integral. That's the exact error. Practically it's difficult to assess so see the above.

On the other hand lower bounds on $P(Y\geq c)$ are considerably more difficult. Large deviations theory gives answers to such questions. There are exist reverse Markov inequalities for positive bounded random variables.