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  "body": "\n## Definition\n\nLet $X, X_1, X_2, \\dots$ be real random variables on one probability space $(\\Omega, \\mathcal{F}, P)$.\nThe sequence $(X_n)$ converges in probability to $X$ if for each $\\varepsilon \u003e 0$,\n$$\\lim_{n \\to \\infty} P\\left(|X_n - X| \u003e \\varepsilon\\right) = 0.$$\nOne writes $X_n \\xrightarrow{P} X$.\n\n## Remarks\n\nThe limit is unique up to equality almost surely.\nIf $X_n \\to X$ in probability, then a subsequence of $(X_n)$ converges to $X$ almost surely.\nThe weak law of large numbers is a result about convergence in probability.\nExample: let $X_n$ be $1$ with probability $1/n$ and $0$ otherwise. Then $X_n \\to 0$ in probability.\n\n## Explanations\n\n### L1 human\n\nConvergence in probability means that, for large $n$, $X_n$ is very likely to be close to $X$.\nIt does not say that one fixed run of the experiment gives a convergent sequence.\n",
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