Multidimensional ergodic theorems and universally representative random sequences (Q1359054)

Let \(\{S_k, k\geq 1\}\) be a sequence of random variables with values in \(\mathbb{Z}^d\), \(d\geq 1\). The sequence \((S_k)\) is said to be universally 2-representative in mean if there exists \(\Omega_0\), \(P(\Omega_0)= 1\), such that for each \(\omega\in\Omega_0\): for every measurable dynamical system \((Y,{\mathcal A},\mu,T)\) and every \(f\in L^2(\mu)\), \({1\over N} \sum^N_{k=1} f\circ T^{S_k(\omega)}\) converges in \(L^2(\mu)\). The main result says that for every i.i.d. sequence \(\{X_k, k\geq 1\}\) with values in \(\mathbb{Z}^d\) such that \(E|X_1|^\delta<\infty\) for some \(\delta>0\), the sequence \(\{X_1+\cdots+ X_k, k\geq 1\}\) is universally 2-representative in mean. In the proof Van der Corput's inequality and some Gaussian methods are used.