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  "body": "\n## Definition\n\nLet $r \\ge 1$ be real. Let $X, X_1, X_2, \\dots$ be real random variables on one probability space with $E[|X|^r] \u003c \\infty$ and $E[|X_n|^r] \u003c \\infty$ for each $n$.\nThe sequence $(X_n)$ converges in $r$-th mean, or in $L^r$, to $X$ if\n$$\\lim_{n \\to \\infty} E\\left[|X_n - X|^r\\right] = 0.$$\nFor $r = 1$ one says convergence in mean, and for $r = 2$ convergence in mean square.\n\n## Remarks\n\nBy [[Markov's inequality|05PWWHYBX5]], $P(|X_n - X| \u003e \\varepsilon) \\le E[|X_n - X|^r] / \\varepsilon^r$. So convergence in $r$-th mean implies convergence in probability.\nFor $s \\ge r \\ge 1$, convergence in $s$-th mean implies convergence in $r$-th mean.\nConvergence in mean does not imply almost sure convergence, and almost sure convergence does not imply convergence in mean.\nExample: 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 $n$.\n\n## Explanations\n\n### L1 human\n\nConvergence in mean asks that the average size of the error $|X_n - X|^r$ goes to $0$.\nMean square convergence, with $r = 2$, is common in statistics, where it says that the mean squared error of an estimator goes to $0$.\n",
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