QUENCHED CENTRAL LIMIT THEOREM FOR DVORETZKY COVERING

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Abstract Let $\{\omega _n\}_{n\geq 1}$ be a sequence of independent and identically distributed random variables on a probability space $(\Omega , \mathcal {F}, \mathbb {P})$ , each uniformly distributed on the unit circle $\mathbb {T}$ , and let $\ell _n=cn^{-\tau }$ for some $c>0$ and $0<\tau <1$ . Let $I_{n}=(\omega _n,\omega _n+\ell _n)$ be the random interval with left endpoint $\omega _n$ and length $\ell _n$ . We study the asymptotic property of the covering time $N_n(x)=\sharp \{1\leq k\leq n: x\in I_k\}$ for each $x\in \mathbb {T}$ . We prove the quenched central limit theorem for the covering time, that is, $\mathbb {P}$ -almost surely, $$ \begin{align*}\frac{N_n(x)-\mathbb{E}_{\mathbb{P}}(N_n(x))}{\sqrt{\sum_{k=1}^n \ell_k(1-\ell_k)}}\end{align*} $$ converges in law to the standard normal distribution.