Donsker's theorem

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In probability theory, Donsker's theorem, named after M. D. Donsker, identifies a certain stochastic process as a limit of empirical processes.

By the classical central limit theorem, for fixed x, the empirical process Gn(x) converges in distribution to a Gaussian (normal) random variable G(x) with mean 0 and variance F(x)(1 − F(x)). Donsker (1952) showed that the sample paths of Gn(x), as functions on the real line R, converge weakly to a stochastic process G in the space \scriptstyle\ell^\infty(\mathbb{R}) of all bounded functions \scriptstyle f:\mathbb{R}\,{\rightarrow}\,\mathbb{R}. The limit process G is a Gaussian process with zero mean and covariance given by

\operatorname{cov}[G(s), G(t)] = E[G(s)G(t)] = F[\min(s, t)] - F(s)F(t). \,

The process G(x) can be written as B(F(x)) where B is a standard Brownian bridge on the unit interval.

[edit] See also

[edit] References

  • M.D. Donsker, "Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems", Annals of Mathematical Statistics, 23:277--281, 1952.