Randomized limit theorems for stationary ergodic random processes and fields (Q6570497)

Consider a \(d\)-dimensional ergodic and homogeneous random field \(X(t)=(X^1(t),\ldots,X^d(t))\) for \(t\in T\), where \(T\) is either \(\mathbb{R}^m\) or \(\mathbb{Z}^m\) for some \(m\geq1\). The authors establish central limit theorems and related results for a randomized version of this process, i.e., for statistics using a finite number of observations of this random field at times chosen randomly and independently of the field itself. These statistics are based on data of the form\N\[\N\left(\sum_{i=1}^{k_n}X^1(\tau_{n,i}^1),\ldots,\sum_{i=1}^{k_n}X^d(\tau_{n,i}^d)\right)\,,\N\]\Nwhere \(\tau_{n,1}^s,\ldots,\tau_{n,{k_n}}^s\) for \(s=1,\ldots,d\) are all independent random vectors, also independent of the random field, such that \(\tau_{n,i}^s\) is chosen uniformly at random from the set \(T_n\), where \(\{T_n\}_n\) is a sequence of Borel sets in \(T\), and where \(k_n\to\infty\) as \(n\to\infty\). These statistics are shown to satisfy a randomized Lindeberg condition. Using this, central limit theorems are established as \(n\to\infty\) for the randomized process which require only second moment conditions, with some corresponding rates of convergence. A corresponding functional central limit theorem is further given, along with a Glivenko-Cantelli theorem and convergence of the empirical processes to the Brownian bridge.