Properties of fourth-order strong mixing rates and its application to random set theory

We shall define the concept of fourth-order strong mixing rates and study their properties. Results are useful for establishing a condition of the form (*) Σ_a,b,c |cum(X_o, X_a, X_b, X_c)| < ∞ or ∫|cum(X_o, X_a, X_b, X_c)| da db dc < ∞ for dependent random variables {X_a}. As an application we shall consider an evaluation of a fourth-order strong mixing rate for a random closed set Z (in the sense of Matheron) and derive the condition (*) for {X_a}, X_a being an outcome of a local measurement upon Z. The result is also applicable to point processes which admit clustering representations.