Subsampling for Random Fields

Suppose {X(t), t ∈ Gd} is a random field in d dimensions, with d ∈ ℤ₊; that is, {X(t), t ∈ Gd} is a collection of random variables X(t) taking values in a state space S, defined on a probability space (Ω, A, P), and indexed by the variable t ∈ Gd. Throughout this chapter, G will stand for either the set of real numbers ℝ, or the set of integers ℤ; thus, the random field {X(t)} is allowed to “run” in either continuous or discrete “time.” Similarly, G₊ will denote ℝ₊ or ℤ₊ (the sets of positive real numbers and integers, respectively) according to whether G = ℝ or G = ℤ.