Perturbed ergodic theorems
A sequence \((p_k)_{k\geq 1}\) of positive integers such that \(\lim_{k\to\infty} p_k= \infty\) is said to be \(L^2\)-satisfying if for any dynamical system \((X, A,\mu, T)\) and any \(f\in L^2(\mu)\) we have \[ \mu\Biggl(x: \lim_{n\to\infty} {1\over n} \sum^n_{k= 1} f(T^{p_k}x)\text{ exists}\Biggr)= 1. \] The main result of the paper is as follows: Let \((\Omega,{\mathcal B},P)\) be a complete probability space on which there is given a sequence \((\theta_k)_{k\geq 1}\) of i.i.d. positive integer-valued random variables such that \(E\theta^\gamma_1< \infty\) for some \(\gamma> 0\). Let \((p_k)_{k\geq 1}\) be a strictly increasing sequence of positive integers such that \(p_k= O(2^{k^\delta})\), \(k\geq 1\), for some \(0< \delta< 1\). Under these assumptions there exists \(\Omega_0\in{\mathcal B}\) with \(P(\Omega_0)= 1\) such that the sequence \((p_k+ \theta_k(\omega))_{k\geq 1}\) is \(L^2\)-satisfying for any \(\omega\in\Omega_0\). The proof makes use of techniques from the harmonic analysis and the theory of Gaussian processes. The result obtained establishes the stability under certain random perturbations of \textit{J. Bourgain's} results [e.g., Isr. J. Math. 61, No. 1, 73-84 (1988; Zbl 0642.28011); ibid. 63, No. 1, 79-97 (1988; Zbl 0677.60042)] concerning the convergence of ergodic averages for certain arithmetic subsequences.
scientific article; zbMATH DE number 1123741