# Donsker's theorem

In probability theory, **Donsker's theorem** (also known as **Donsker's invariance principle**, or the **functional central limit theorem**), named after Monroe D. Donsker, is a functional extension of the central limit theorem.

Let be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let . The stochastic process is known as a random walk. Define the diffusively rescaled random walk by

The central limit theorem asserts that converges in distribution to a standard Gaussian random variable as . Donsker's invariance principle^{[1]}^{[2]} extends this convergence to the whole function . More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space , the random function converges in distribution to a standard Brownian motion as

## History

Let *F*_{n} be the empirical distribution function of the sequence of i.i.d. random variables with distribution function *F.* Define the centered and scaled version of *F*_{n} by

indexed by *x* ∈ **R**. 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.

**Theorem** (Donsker, Skorokhod, Kolmogorov) The sequence of *G*_{n}(*x*), as random elements of the Skorokhod space , converges in distribution to a Gaussian process *G* with zero mean and covariance given by

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

Kolmogorov (1933) showed that when *F* is continuous, the supremum and supremum of absolute value, 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.^{[3]}

In 1952 Donsker stated and proved (not quite correctly)^{[4]} 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].

^{[2]}

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 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^{[4]} that there exist *X _{i}*, iid uniform in [0,1] and a sequence of sample-continuous Brownian bridges

*B*

_{n}, such that

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

## References

- ↑ Donsker, M.D. (1951). "An invariance principle for certain probability limit theorems".
*Memoirs of the American Mathematical Society, 1951, no. 6*. - ↑
^{2.0}^{2.1}Donsker, M. D. (1952). "Justification and extension of Doob's heuristic approach to the Kolmogorov–Smirnov theorems".*Annals of Mathematical Statistics*.**23**: 277–281. MR 47288. Zbl 0046.35103. doi:10.1214/aoms/1177729445. - ↑ Doob, Joseph L. (1949). "Heuristic approach to the Kolmogorov–Smirnov theorems".
*Annals of Mathematical Statistics*.**20**: 393–403. MR 30732. Zbl 0035.08901. doi:10.1214/aoms/1177729991. - ↑
^{4.0}^{4.1}Dudley, R.M. (1999).*Uniform Central Limit Theorems*. Cambridge University Press. ISBN 0-521-46102-2.