The Jensen gap $\mathbb{E}[|\overline X|] - |\mu|$

Question

Let $ X_1, X_2, \dots, X_n $ be a sequence of i.i.d. random variables with finite mean $ \mu $ and variance $ \sigma^2 $. Let $ \overline X = \frac{1}{n}\sum_{i=1}^n X_i $ denote the sample average. I would like to bound the Jensen gap $ \mathbb{E}[|\overline X|] - |\mu| $ for $ \mu \neq 0 $. Simulations suggest that the gap decays exponentially in $ n $, which indeed can be verified asymptotically by the central limit theorem. But I would like an analogous bound that is valid for all $ n $. Any help would be much appreciated!

Answer 1

By the reverse triangle inequality and Jensen's inequality: $$ |E(|\bar X| - |\mu|)| \le E(|\bar X - \mu|) = E\left(\sqrt{(\bar X - \mu)^2}\right) \le \sqrt{\operatorname{Var}(\bar X)} = \frac{\sigma}{\sqrt n} $$ where $\sigma$ is the standard deviation of $X_1$.

I suspect it is a misstep to assume that the asymptotic you get from a normal distribution will be representative of what happens in general. The asymptotic for $E(|\bar X|)$ might depend on information in the tail behavior of the $X_i$'s, which is information you lose when you apply the CLT heuristic. The following R code suggests that you get polynomial decay when $X_i$ has a $t$-distribution:

mysim <- function(n, reps) {
  moo <- matrix(rt(n = n * reps, df = 4, ncp = 0), nrow = n)
  return(mean(abs(colMeans(moo))))
}

q <- sapply(100:200, function(n) mysim(n, 10000))
plot(q)