Nonsingular group actions and stationary S𝛼S random fields

Abstract

This paper deals with measurable stationary symmetric stable random fields indexed by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and their relationship with the ergodic theory of nonsingular <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -actions. Based on the phenomenal work of Rosiński (2000), we establish extensions of some structure results of stationary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S alpha upper S"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi> α </mml:mi> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">S\alpha S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> processes to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S alpha upper S"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi> α </mml:mi> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">S\alpha S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fields. Depending on the ergodic theoretical nature of the underlying action, we observe different behaviors of the extremes of the field.