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  "body": "\n## Summary\n\nA family of [[probability measures|059TJN0NZ1]] on a topological space is tight if all of them live on one [[compact set|0R6EWFKRCF]] up to a small mass.\nThe mass left out can be made as small as one likes.\nTightness rules out mass escaping to infinity.\nBy [[Prokhorov's theorem|D6FMEQYA76]], on a [[Polish space|584GXXY73M]] a family is tight if and only if every sequence in it has a [[weakly convergent|MBW9JCGZC1]] subsequence.\nSo proofs of [[convergence in distribution|0SRYJGXKR4]] often have two steps: prove tightness, then identify the limit.\n\n## Explanations\n\n### L1 human\n\nThink of each measure as a pile of sand of total mass $1$ spread on the line.\nThe family is tight if one fixed bounded interval holds almost all the sand of every pile, say all but $1\\%$, and the same works for any small percentage.\nThe normal laws with means between $-1$ and $1$ form a tight family.\nThe point masses at $1, 2, 3, \\dots$ do not: the sand runs off to infinity, and no single interval catches it.\n",
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