Convergence in mean

Definition

Let $r \ge 1$ be real. Let $X, X_1, X_2, \dots$ be real random variables on one probability space with $E[|X|^r] < \infty$ and $E[|X_n|^r] < \infty$ for each $n$. The sequence $(X_n)$ converges in $r$-th mean, or in $L^r$, to $X$ if $$\lim_{n \to \infty} E\left[|X_n - X|^r\right] = 0.$$ For $r = 1$ one says convergence in mean, and for $r = 2$ convergence in mean square.

Remarks

By Markov's inequality, $P(|X_n - X| > \varepsilon) \le E[|X_n - X|^r] / \varepsilon^r$. So convergence in $r$-th mean implies convergence in probability. For $s \ge r \ge 1$, convergence in $s$-th mean implies convergence in $r$-th mean. Convergence in mean does not imply almost sure convergence, and almost sure convergence does not imply convergence in mean. Example: on $[0, 1]$ with the uniform measure, $X_n = n \, 1_{[0, 1/n]}$ converges to $0$ almost surely but $E[X_n] = 1$ for all $n$.

Explanations

L1 human

Convergence in mean asks that the average size of the error $|X_n - X|^r$ goes to $0$. Mean square convergence, with $r = 2$, is common in statistics, where it says that the mean squared error of an estimator goes to $0$.