  We consider "randomized" statistics constructed by using a finite number of
observations a random field at randomly chosen points. We generalize the
invariance principle (the functional CLT), the Glivenko--Cantelli theorem, the
theorem about convergence to the Brownian bridge and the Kolmogorov theorem
about the limit distribution of the empirical distribution function, as well as
an improved version of the CLT in A. Tempelman, Randomized multivariate central
limit theorems for ergodic homogeneous random fields, Stochastic Processes and
their Applications.
  143 (2022), 89-105. The randomized approach, introduced in the mentioned
work, allows to extend these theorems to all ergodic homogeneous random fields
on $\Z^m$ and $\R^m.$
