A New Class of Second Order Self-Similar Processes A. Gefferth , D. Veitch , I. Ruzsa , I. Maricza , S. Moln´ ar December 19, 2003 High Speed Networks Laboratory, Department of Telecommunications and Media Informatics Budapest University of Technology and Economics Magyar Tud´ osok k¨ or´ utja 2, H-1117 Budapest, Hungary ARC Special Research Center for Ultra-Broadband Information Networks Department of Electrical and Electronic Engineering The University of Melbourne, Australia. Alfr´ ed R´ enyi Institute of Mathematics Hungarian Academy of Sciences POB 127, H-1364 Budapest, Hungary Corresponding author: D. Veitch Abstract Self-similarity in discrete second-order stationary processes is defined as a fixed point of a renormalisation operator consisting of aggregation normalised by the variance, rather than by the traditional power-law factor. This broader definition reveals a new class of self-similar processes. Keywords: self-similarity, renormalisation, fractional noise, stationarity, second-order. 1 Introduction In second-order stationary processes the second order properties are fully described by a single deterministic quantity, the covariance function. Its normalised form, the correlation function, plays the role of the ‘shape parameter’ of the process. This paper concerns the definition and correlation structure of self-similar discrete second-order stationary processes. In this context self-similarity is seen as invariance of the correlation function under a suitable kind of renormalisation operation. Supported by the Australian-European Awards Program, an Australian Government funded Award administered by the Department of Employment, Education, Training and Youth Affairs. Corresponding Author. Australian Research Council Special Research Center, Department of Electrical & Elec- tronic Engineering, University of Melbourne Victoria 3010 Australiai, Supported by Ericsson Australia Supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant Nos. 25617, 29759, 38396. 1