Part CCSIR NET December 2023kolmogorov-scaling

Kolmogorov scaling

Let $X_{1}, \dots, X_{n}$ be a random sample from an unknown absolutely continuous CDF F, and $F_{0} a$ specified absolutely continuous CDF. For $H_{0} : F = F_{0}$ vs $H_{1} : F \neq = F_{0}$ , consider $T_{1, n}$ = sup_x | $F_{n} (x) - F_{0} (x)$ | and $T_{2, n}$ = n·sup_x | $F_{n} (x) - F_{0} (x)$ |, where $F_{n}$ is the empirical CDF. Which of the following statements are true?

A. $T_{1, n}$ → 0 in probability as $n \to \infty$ under $H_{0}$
B. $T_{2, n}$ → 0 in probability as $n \to \infty$ under $H_{0}$
C.lim P_F( $T_{2, n}$ > 1) = 1 for all F
D. $T_{2, n}$ converges in distribution to a degenerate random variable under $H_{0}$

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