# Question regarding asymptotic assumptions and hypotesis testing paradox for large samples

Suppose we would like to verify if a r.v $$X$$ follows a distribution with cumulative density as $$F$$, if $$n$$ goes to $$\infty$$ I'm able to use komogorov test which states reject $$H_0$$ (stating that $$X$$ does follow the such density) when:

$$\sqrt{n}.sup|F(x)-F_{emp}(x)|>K_{\alpha}$$

where $$K_{\alpha}$$ is some quantile from komogorov distribution and $$F_{emp}$$ as empiric cumulative distribution

But wikipedia states that: That's komogorov test will ever reject $$H_0$$ when $$n \rightarrow \infty$$ as $$power=P(Reject \hspace{0.1cm} H_0|\theta)$$ where $$\theta$$ is the 'maximum' (supreme) distance between empiric cumulative and $$F$$.

So here's my question:

Why is this test even useful? since it'll reject $$H_0$$ when $$n\rightarrow \infty$$ and the test holds only if $$n \rightarrow \infty$$.

PS: It's a specific instance from this problem there is so many more tests which lead this paradox.

• The concern is the test just works if $n\rightarrow \infty$ but when it occurs so power=1, that's I'll always reject $H_0$ whatever cumulative I'm comparing then what is the test for? Dec 7, 2021 at 2:03