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Random walks driven by low moment measures. (English) Zbl 1262.60005

Random walks on a locally compact unimodular group \(G\) are considered. The authors investigate the rate of decay of the probability to return to a neighborhood of the origin (identity element \(e\)) at the \(2n\)-th step. This probability corresponds to the value at \(e\) of the density \(\varphi^{(2n)}\) of \(2n\)-fold convolution of the one-step density \(\varphi\) (w.r.t. the Haar measure). It is assumed that \(\varphi\) belongs to the class \(S_{G,\rho}\) of symmetric continuous densities with finite \(\int\rho\varphi\) for some fixed moment function \(\rho: G\to [0,\infty)\).
For different types of moment functions \(\rho\) and groups \(G\), the fastest possible rates of decay \[ \Phi_{G,\rho}(n)=\inf\{\varphi^{(2n)}(e)\;:\;\varphi\in S_{G,\rho}\} \] are derived, e.g., for polycyclic groups with exponential volume growth and \(\rho(g)=\exp(c[\log(1+|g|]^\alpha)\) (\(|g|\) being a natural word length measuring the distance from \(g\) to \(e\)), \[ \log \Phi_{G,\rho}(n)\approx n\exp(-c_1[\log n]^\alpha). \]

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G50 Sums of independent random variables; random walks
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