Moderate and small deviations for the ranges of one-dimensional random walks
Let \(S(n):= X(1)+\cdots+ X(n)\) \((n\geq 1)\) \((S(0):= 0)\) be an integer valued random walk (here, \(X(1),X(2),\dots\) is an i.i.d. integer valued sequence) It is assumed that \(E[X(1)]= 0\) and \(\sigma^2= E[X^2(1)]< \infty\). Furthermore, it is assumed that \(Z\) is the smallest group supporting \(S(0),S(1),\dots\)\ . For any \(\Delta\subset \mathbb R^+\) let \(S(\Delta):= \{S(k): k\in\Delta\}\). Let \((S_1(n))_{n\geq 0},\dots, (S_p(n))_{n\geq 0}\) be independent copies of \((S(n))_{n\geq 0}\). In the sequel \((b_n)\) denotes a positive sequence such that \(b_n\to\infty\) and \(b_n =o(n)\) as \(n\to\infty\). The main results are the following theorems. Theorem 1. For any \(\lambda> 0\) and integer \(p\geq 1\), \[ {1\over b_n}\log P(\#(S_1([0, n])\cap\cdots\cap S_p([0, n]))\geq \lambda\sqrt{nb_n})\to -{p\lambda^2\over 2\sigma^2}. \] Theorem 2. For any \(\lambda> 0\), \[ {1\over b_n}\log P(\# S([0, n])\leq \lambda\sqrt{n/b_n})\to -{\pi^2\sigma^2\over 2\lambda^2}. \] Theorem 3. For any integer \(p\geq 1\), \[ \limsup_{n\to\infty} {1\over\sqrt{n\log\log n}}\#(S_1([0, n])\cap\cdots\cap S_p([0, n]))= \sqrt{{2\sigma^2\over p}}\text{ a.s.} \] and, if \(p= 1\), \[ \liminf_{n\to\infty} \sqrt{{\log\log n\over n}}\# S([0, n])= {\pi\sigma\over \sqrt{2}}\text{ a.s.} \] A recent paper by \textit{A. Klenke} and \textit{P. Mörters} [Ann. Probab. 33, No. 4, 1255--1301 (2005; Zbl 1080.60078)] on the lower tail of intersections of Brownian paths suggests that the small deviation result in Theorem 2 (exponential decay) does not hold for the intersection of ranges.
scientific article; zbMATH DE number 5126786
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