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| In [[probability theory]], '''Donsker's theorem''', named after [[Monroe D. Donsker]], identifies a certain [[stochastic process]] as a limit of [[empirical process]]es. It is sometimes called the '''functional central limit theorem'''.
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| A centered and scaled version of [[empirical distribution function]] ''F''<sub>''n''</sub> defines an [[empirical process]]
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| : <math> G_n(x)= \sqrt n ( F_n(x) - F(x) ) \, </math>
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| indexed by ''x'' ∈ '''R'''.
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| '''Theorem''' (Donsker, Skorokhod, Kolmogorov) The sequence of ''G''<sub>''n''</sub>(''x''), as random elements of the [[Skorokhod space]] <math>\mathcal{D}(-\infty,\infty)</math>, [[convergence in distribution|converges in distribution]] to a [[Gaussian process]] ''G'' with zero mean and covariance given by
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| : <math>\operatorname{cov}[G(s), G(t)] = E[G(s) G(t)] = \min\{F(s), F(t)\} - F(s)F(t). \,</math>
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| The process ''G''(''x'') can be written as ''B''(''F''(''x'')) where ''B'' is a standard [[Brownian bridge]] on the unit interval.
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| ==History==
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| By the classical [[central limit theorem]], for fixed ''x'', the random variable ''G''<sub>''n''</sub>(''x'') [[converges in distribution]] to a [[normal distribution|Gaussian (normal)]] [[random variable]] ''G''(''x'') with zero mean and variance ''F''(''x'')(1 − ''F''(''x'')) as the sample size ''n'' grows.
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| Kolmogorov (1933) showed that when ''F'' is [[continuous function|continuous]], the supremum <math>\scriptstyle\sup_t G_n(t)</math> and supremum of absolute value, <math>\scriptstyle\sup_t |G_n(t)|</math> [[convergence in distribution|converges in distribution]] to the laws of the same functionals of the [[Brownian bridge]] ''B''(''t''), see the [[Kolmogorov–Smirnov test]]. In 1949 Doob asked whether the convergence in distribution held for more general functionals, thus formulating a problem of [[weak convergence of measures|weak convergence]] of random functions in a suitable [[function space]].<ref>{{cite journal
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| |first=Joseph L. |last=Doob|authorlink=Joseph L. Doob
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| |title=Heuristic approach to the Kolmogorov–Smirnov theorems
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| |journal=[[Annals of Mathematical Statistics]]
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| |volume=20 |issue= |pages=393–403 |year=1949
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| |doi=10.1214/aoms/1177729991 |mr=30732 | zbl = 0035.08901
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| }}</ref>
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| In 1952 Donsker stated and proved (not quite correctly)<ref name="dudley1999">{{cite book
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| |first=R.M. |last=Dudley|authorlink=Richard M. Dudley
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| |title=Uniform Central Limit Theorems
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| |publisher=Cambridge University Press
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| |year=1999
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| |isbn=0-521-46102-2
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| }}</ref> a general extension for the Doob-Kolmogorov heuristic approach. In the original paper, Donsker proved that the convergence in law of ''G<sub>n</sub>'' to the Brownian bridge holds for [[uniform distribution (continuous)|Uniform[0,1]]] distributions with respect to uniform convergence in ''t'' over the interval [0,1].<ref>{{cite journal
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| |first=M. D. |last=Donsker |authorlink=Monroe D. Donsker
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| |title=Justification and extension of Doob's heuristic approach to the Kolmogorov–Smirnov theorems
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| |journal=[[Annals of Mathematical Statistics]]
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| |volume=23 |issue= |pages=277–281 |year=1952
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| |doi=10.1214/aoms/1177729445 |mr=47288 | zbl = 0046.35103
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| }}</ref>
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| However Donsker's formulation was not quite correct because of the problem of measurability of the functionals of discontinuous processes. In 1956 Skorokhod and Kolmogorov defined a separable metric ''d'', called the ''Skorokhod metric'', on the space of [[cadlag function]]s on [0,1], such that convergence for ''d'' to a continuous function is equivalent to convergence for the sup norm, and showed that ''G<sub>n</sub>'' converges in law in <math>\mathcal{D}[0,1]</math> to the Brownian bridge.
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| Later Dudley reformulated Donsker's result to avoid the problem of measurability and the need of the Skorokhod metric. One can prove<ref name="dudley1999" /> that there exist ''X<sub>i</sub>'', iid uniform in [0,1] and a sequence of sample-continuous Brownian bridges ''B''<sub>''n''</sub>, such that
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| :<math>\|G_n-B_n\|_\infty</math>
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| is measurable and [[convergence in probability|converges in probability]] to 0. An improved version of this result, providing more detail on the rate of convergence, is the [[Komlós–Major–Tusnády approximation]].
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| ==See also== | |
| *[[Glivenko–Cantelli theorem]]
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| *[[Kolmogorov–Smirnov test]]
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| == References ==
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| {{reflist}}
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| {{DEFAULTSORT:Donsker's Theorem}}
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| [[Category:Probability theorems]]
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| [[Category:Statistical theorems]]
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| [[Category:Empirical process]]
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