# Kolmogorov Distribution D statistics

As far as I have searched the cumulative distribution function of 𝐾, asymptotically (kolmogorov distribution) is given by Pr(𝐾≤𝑥)=1−2∑∞𝑘=1(−1)𝑘−1𝑒−2𝑘2𝑥2=2𝜋√𝑥∑∞𝑘=1𝑒−(2𝑘−1)2𝜋2/(8𝑥2).

But I can not obtain the proof of it. Could you provide me with the proof or share some documents, links about it?

• Welcome to the site. It would help if you could format your question in LaTeX. – Peter Flom Mar 28 at 10:59

Let $$Y_1, \ldots, Y_n \sim_{\text{iid}} dF$$ and $$\hat{F}_n$$ the corresponding empirical cumulative distribution.
The Kolmogorov statistic for the assumption $$F = G$$ is $$K_n = \sup |\hat{F}_n(x) - G(x)|$$.
Under the null hypothesis $$F=G$$, one has $$\Pr\left(K_n > \frac{c}{\sqrt{n}}\right) \to 2 \sum_{k=1}^\infty (-1)^{k-1} \exp(-2k^2c^2).$$