Limit theorems for the ratio of the empirical distribution function to the true distribution function

摘要: We consider almost sure limit theorems for $$\begin{gathered} \parallel \Gamma _n /I\parallel _{ a_{ n} }^{ 1} \equiv \sup (\Gamma (t)/t) \hfill \\ \leqq t 1 \end{gathered} $$ and I/\Gamma = (t/\Gamma (t)) whereΓ n is the empirical distribution function of a random sample ofn uniform (0, 1) variables anda ↓0. It shown that (1) ifna /log2 n→∞ then both $$\parallel converge to a.s.; (2) /log2 n=d>0 (d>1) then ( )$$ has an surely finite superior which solution certain transcendental equation; and (3) /log2 n→0 have +∞ surely. Similar results are established for inverse functionΓ −1 .