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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?

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  • $\begingroup$ Welcome to the site. It would help if you could format your question in LaTeX. $\endgroup$ – Peter Flom Mar 28 at 10:59
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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 $$ \Pr\left(K_n > \frac{c}{\sqrt{n}}\right) \to 2 \sum_{k=1}^\infty (-1)^{k-1} \exp(-2k^2c^2). $$

This result is proved in Billingsley' book Convergence of probability measures. The proof involves the Brownian motion.

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