Donskerin

Donskerin is a term used in probability theory and statistics. It refers to a class of stochastic processes that arise from the empirical distribution function. Specifically, Donsker's theorem, also known as the functional central limit theorem, states that under certain conditions, the empirical distribution function, when properly normalized and viewed as a stochastic process, converges weakly to a Brownian bridge. This means that for large sample sizes, the behavior of the scaled empirical process resembles that of a Brownian bridge.

The empirical distribution function, denoted by $F_n(x)$, is a step function that approximates the true cumulative distribution function of a random variable based on a sample. For a sample $X_1, X_2, \ldots, X_n$ of independent and identically distributed random variables with cumulative distribution function $F(x)$, the empirical distribution function is defined as $F_n(x) = \frac{1}{n} \sum_{i=1}^n \mathbf{1}_{X_i \leq x}$, where $\mathbf{1}$ is the indicator function.

The normalized empirical process is then defined as $\sqrt{n}(F_n(x) - F(x))$. Donsker's theorem states that this process converges weakly to a Gaussian process. If $F$ is the uniform distribution on $[0,1]$, this limiting process is a Brownian bridge. The concept of Donsker classes is a generalization of this idea to more complex function spaces. A Donsker class of functions is a collection of functions for which a similar functional central limit theorem holds. These results are fundamental in non-parametric statistics and the theory of empirical processes, providing a theoretical basis for various statistical tests and estimators.