Logarithmic averages for the local times of recurrent random walks and Lévy processes
Let \(X_n\) be a recurrent random walk in \(\mathbb{Z}^d\) in the domain of attraction of a nondegenerate strictly stable random variable \(U\) of index \(\beta\). Then \(X_n/ b(n)\to U\) in law where \(b(n)\) is regularly varying of order \(1/\beta\). Here \(b(x)\) can always be taken as a continuous and monotone increasing function with \(b(0) =0\). With the notation \(p_n (x)= P(X_n =x)\) then \(p_n (0) \sim b^{-d} (n)\) is regularly varying of order \(-d/ \beta\), and \(g(n)= \sum^n_{j=1} p_j (0) \sim \int^n_1 b^{-d} (t) dt\) is regularly varying of order \(1- d/\beta\). Further let \(L^x_n=\{\text{number of }j\mid X_j=x\), \(1\leq j\leq n\}\), \(L_n= L^0_n\). The authors [Ann. Probab., to appear] have shown the law of the iterated logarithm \[ \limsup_{n\to \infty} L_n/ (g(n/ \log \log g(n)) \log \log g(n))= a_0 \quad \text{a.s.}, \] where \(a_0\) is a computable constant. By taking logarithmic averages the authors go from the \(\limsup\) to the true limit theorems \[ \lim_{N\to \infty} (1/\log g(N)) \sum^N_{n=1} p_n (0) L_n/ g^2 (n)= 1 \quad \text{a.s.} \] and \[ \lim_{N\to \infty} (\log g(N))^{- 1/2} \sum^N_{n=1} p_n (0) (L^0_n- L^x_n)= 4a (x) N(0, 1) \] in a weak sense for symmetric \(X_n\) and \(a^2 (x)= \sum^\infty_{n=1} (p_n (0)- p_n (x))\). Besides logarithmic averages \(f\)-averages are considered, in which \(\log g(N)\) is replaced by \(f(g (N))\), and \(1/g^2 (n)\) is replaced by \(-f'' (g(n))\).
scientific article; zbMATH DE number 828749