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  "body": "\n## Summary\n\nA sequence of random variables can converge to a limit in several different senses.\nThe main modes are [[almost sure convergence|04P7D63W8P]], [[convergence in probability|0EFDDXTVD6]], [[convergence in mean|0TTA57SEJG]] and [[convergence in distribution|0JDAQHWA9N]].\nAlmost sure convergence and convergence in mean each imply convergence in probability, and convergence in probability implies convergence in distribution. The converse implications are false in general.\n\n## Remarks\n\nConvergence in distribution concerns only the laws of the random variables. So the random variables can be on different probability spaces.\nThe law of large numbers is a statement about convergence in probability or almost surely. The central limit theorem is a statement about convergence in distribution.\n\n## Explanations\n\n### L1 human\n\nFor numbers, there is one meaning of \"$x_n \\to x$\". For random variables, there are several, because a random variable has many values at once.\nSome modes ask that the values themselves come close, and the weakest mode asks only that the probabilities come close.\n",
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