Summary
We consider almost sure limit theorems for\(\begin{gathered} \parallel \Gamma _n /I\parallel _{ a_{ n} }^{ 1} \equiv \sup (\Gamma _n (t)/t) \hfill \\ a_{ n} \leqq t \leqq 1 \hfill \\ \end{gathered} \) and\(\begin{gathered} \parallel I/\Gamma _n \parallel _{ a_{ n} }^{ 1} = \sup (t/\Gamma _n (t)) \hfill \\ a_{ n} \leqq t \leqq 1 \hfill \\ \end{gathered} \) whereΓ n is the empirical distribution function of a random sample ofn uniform (0, 1) random variables anda n ↓0. It is shown that (1) ifna n /log2 n→∞ then both\(\parallel \Gamma _n /I\parallel _{ a_{ n} }^{ 1} \) and\(\parallel I/\Gamma _n \parallel _{ a_{ n} }^{ 1} \) converge to 1 a.s.; (2) ifna n /log2 n=d>0 (d>1) then\(\parallel \Gamma _n /I\parallel _{ a_{ n} }^{ 1} ( \parallel I/\Gamma _n \parallel _{ a_{ n} }^{ 1} )\) has an almost surely finite limit superior which is the solution of a certain transcendental equation; and (3) ifna n /log2 n→0 then\(\parallel \Gamma _n /I\parallel _{ a_{ n} }^{ 1} \) and\(\parallel I/\Gamma _n \parallel _{ a_{ n} }^{ 1} \) have limit superior +∞ almost surely. Similar results are established for the inverse functionΓ −1 n .
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Wellner, J.A. Limit theorems for the ratio of the empirical distribution function to the true distribution function. Z. Wahrscheinlichkeitstheorie verw Gebiete 45, 73–88 (1978). https://doi.org/10.1007/BF00635964
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DOI: https://doi.org/10.1007/BF00635964