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 (*) [Sigma]a,b,c cum(Xo, Xa, Xb, Xc) < [infinity] or [integral operator]cum(Xo, Xa, Xb, Xc) da db dc < [infinity] for dependent random variables {Xa}. 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 {Xa}, Xa being an outcome of a local measurement upon Z. The result is also applicable to point processes which admit clustering representations.