Self-Similarity and Lamperti Transformation for Random Fields

Abstract

We define multi-self-similar random fields, that is, random fields that are self-similar component-wise. We characterize them, relate them to stationary random fields using a Lamperti-type transformation and study these stationary fields. We also extend the notions of local stationarity and local stationarity reducibility to random fields. Our work is motivated by applications arising from climatological and environmental sciences. We illustrate these new concepts with the fractional Brownian sheet and the Lévy fractional Brownian random field.

Keywords:

• Fractional Brownian sheet
• Lamperti transformation
• Lévy fractional Brownian random field
• Local stationarity
• Random field
• Reducibility
• Self-similarity

AMS Subject Classification:

• 60G18
• 62M10

ACKNOWLEDGMENTS

The authors thank the editors and referees for constructive suggestions that have improved the content and presentation of this article. This research was started while Marc G. Genton was visiting the University of Toulouse 1, and continued while Olivier Perrin was visiting North Carolina State University and Murad S. Taqqu was visiting the University of North Carolina at Chapel Hill and the Statistical and Applied Mathematical Sciences Institute (SAMSI) at the Research Triangle Park, NC. Olivier Perrin also visited Texas A&M University to work on this project. This research was partially supported by NSF grants DMS-0102410 at Boston University, and DMS-0504896 and CMG ATM-0620624 at Texas A&M University.