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On the covering time of a disc by simple random walk in two dimensions. (English) Zbl 0789.60019
Cinlar, E. (ed.) et al., Seminar on stochastic processes, 1992. Held at the Univ. of Washington, DC, USA, March 26-28, 1992. Basel: Birkhäuser. Prog. Probab. 33, 189-207 (1992).
Let \(\{S(j)\}\) be the simple random walk on the plane and let \(T_ n=\inf \{j:B_ n \subset S[0,j] \}\), where \(B_ n=\{z:z \in \mathbb{Z}^ 2,| z |^ 2<n\}\) and \(S[0,j]= \{S(i) : 0 \leq i \leq j\}\). The author investigates the limit distribution of \(T_ n\) and proves \[ e^{-4/t} =\liminf_{n \to \infty} \mathbb{P} \{\log T_ n \leq t(\log n)^ 2\} \leq \limsup_{n \to \infty} \mathbb{P} \{\log T_ n \leq t(\log n)^ 2\} \leq e^{-2/t}. \]
For the entire collection see [Zbl 0780.00020].
Reviewer: P.Révész (Wien)

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
60D05 Geometric probability and stochastic geometry