Randomized consistent statistical inference for random processes and fields

We propose a randomized approach to the consistent statistical analysis of random processes and fields on $${\mathbb {R}}^m$$ R m and $${\mathbb {Z}}^m, m=1,2,...$$ Z m , m = 1 , 2 , . . . , which is valid in the case of strong dependence: the parameter of interest $$\theta $$ θ only has to possesses a consistent sequence of estimators $${\hat{\theta }}_n$$ θ ^ n . The limit theorem is related to consistent sequences of randomized estimators $${\hat{\theta }}_n^*$$ θ ^ n ∗ ; it is used to construct consistent asymptotically efficient sequences of confidence intervals and tests of hypotheses related to the parameter $$\theta $$ θ . Upper bounds for “admissible” sequences of normalizing coefficients in the limit theorem are established for some statistical models in Part 2.