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Functional laws of the iterated logarithm for local times of recurrent random walks on \(\mathbb{Z}^2\). (English) Zbl 0913.60052

Let \(X_n\) be a symmetric random walk on the planar integer lattice. Let \(L^0_n\) denote the number of times that \(X_k= 0\) among \(1\leq k\leq n\). Theorem 1 of this paper proves functional (in \(x\)) laws of the iterated logarithm for scaled forms of \(L^0_{t(n,x)}\), i.e. for \(L^0_{t(n,x)}/g(n)\log g(n)\). Here \(g(\cdot)\) is the linear interpolation of \(g(n)= \sum^n_{k= 0}P[X_{n+ 1}- X_n= 0]\) (assumed slowly varying at \(\infty\)) and for each \(0\leq x\leq 1\), \(\lim_{n\to\infty} g(t(n, x))/g(n)= x\). The limit set comprises non-decreasing right-continuous functions \(m\) on \([0,1]\) such that \(m(0)= 0\) and \(\int^1_0 x^{-1}dm(x)\leq 1\). This result is contrasted with Theorem 2 where \(t(n,x)= nx\). In this case \(L^0_{t(n,x)}/g(n)\log \log g(n)\) has a different limit set consisting of functions of the form \(m_1+ (m_2- m_1)I[x> \tau]\) for suitable constants \(m_i\) and \(\tau\).

MSC:

60G50 Sums of independent random variables; random walks
60F17 Functional limit theorems; invariance principles
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