Donsker's theorem

In probability theory, Donsker's theorem, named after M. D. Donsker, identifies a certain stochastic process as a limit of empirical processes. It is sometimes called the functional central limit theorem.

A centered and scaled version of empirical distribution function F_n defines an empirical process

$G_n(x)= \sqrt n ( F_n(x) - F(x) ) \,$

indexed by x ∈ R.

Theorem (Donsker, Skorokhod, Kolmogorov) The sequence of G_n(x), as random elements of the Skorokhod space $\mathcal{D}(-\infty,\infty)$, converges in distribution to a Gaussian process G 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.

History

By the classical central limit theorem, for fixed x, the random variable G_n(x) converges in distribution to a Gaussian (normal) random variable G(x) with zero mean and variance F(x)(1 − F(x)) as the sample size n grows.

Kolmogorov (1933) showed that when F is continuous, the supremum $\scriptstyle\sup_t G_n(t)$ and supremum of absolute value, $\scriptstyle\sup_t |G_n(t)|$ 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 random functions in a suitable function space.^[1]

In 1952 Donsker stated and proved (not quite correctly)^[2] a general extension for the Doob-Kolmogorov heuristic approach. In the original paper, Donsker proved that the convergence in law of G_n to the Brownian bridge holds for Uniform[0,1] distributions with respect to uniform convergence in t over the interval [0,1].^[3]

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 functions on [0,1], such that convergence for d to a continuous function is equivalent to convergence for the sup norm, and showed that G_n converges in law in $\mathcal{D}[0,1]$ to the Brownian bridge.

Later Dudley reformulated Donsker's result to avoid the problem of measurability and the need of the Skorokhod metric. One can prove^[2] that there exist X_i, iid uniform in [0,1] and a sequence of sample-continuous Brownian bridges B_n, such that

$\|G_n-B_n\|_\infty$

is measurable and 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.

See also

• Glivenko–Cantelli theorem
• Kolmogorov–Smirnov test

References

1. ^ Doob, J.L. (1949). "Heuristic approach to the Kolmogorov–Smirnov theorems". Annals of Mathematical Statistics 20: 393–403. doi:10.1214/aoms/1177729991. MR30732. Zbl 0035.08901. 
2. ^ ^{a b} Dudley, R.M. (1999). Uniform Central Limit Theorems. Cambridge University Press. ISBN 0 521 46102 2. 
3. ^ Donsker, M.D. (1952). "Justification and extension of Doob's heuristic approach to the Kolmogorov–Smirnov theorems". Annals of Mathematical Statistics 23: 277–281. doi:10.1214/aoms/1177729445. MR47288. Zbl 0046.35103.