Chebyshev’s Inequality For Bounding From Below? (12/365)

Problem 6.2 asks the following. Let f = f(x) be a nonnegative even function that is nondecreasing for positive x. Then for a random variable \theta with |\theta(\omega)| \le C,

\displaystyle  \frac{Ef(\theta) - f(\epsilon)}{f(C)} \le P(|\theta - E\theta | \ge \epsilon) \le \frac{Ef(\theta - E\theta)}{f(\epsilon)}

From the looks of it, the upper bound seems simple enough because it looks like a direct application of Chebyshev’s inequality. Whereas, the lower bound doesn’t look familiar. If we go back to the proof of Chebyshev’s inequality we can try to see if we can arrive at a lower bound instead of an upper bound. It happens that we can once we know that the random variable is bounded |\theta(\omega)| \le C.

\displaystyle  P(\theta \ge \epsilon) = EI(\theta \ge \epsilon) \\ \ge E\frac{\theta}{C} I(\theta \ge \epsilon) \\ = E\frac{\theta}{C} - E\frac{\theta}{C}I(\theta < \epsilon) \\ \ge E\frac{\theta}{C} - E\frac{\epsilon}{C}I(\theta < \epsilon) \\ \ge \frac{E(\theta - \epsilon)}{C}

Applying this quickly leads to the required solution. The problem points out the case where f(x) = x^2, which leads to

\displaystyle  \frac{E\theta^2 - \epsilon^2}{C^2} \le P( | \theta - E\theta | \ge \epsilon) \le \frac{V\theta}{\epsilon^2}

We are in essence bounding the probability that the random variable \theta takes on a value close to its mean E\theta, which I should imagine is pretty useful to have not so much for computing the probabilities but for asymptotic analysis like we used for the law of large numbers.

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