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March 2017 Behavior of the generalized Rosenblatt process at extreme critical exponent values
Shuyang Bai, Murad S. Taqqu
Ann. Probab. 45(2): 1278-1324 (March 2017). DOI: 10.1214/15-AOP1087


The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches the corners of the triangle, because the limit process will depend on how these corners are approached. All limits are in the sense of weak convergence in $C[0,1]$. These limits cannot be strengthened to convergence in $L^{2}(\Omega)$.


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Shuyang Bai. Murad S. Taqqu. "Behavior of the generalized Rosenblatt process at extreme critical exponent values." Ann. Probab. 45 (2) 1278 - 1324, March 2017.


Received: 1 September 2014; Revised: 1 May 2015; Published: March 2017
First available in Project Euclid: 31 March 2017

zbMATH: 06797092
MathSciNet: MR3630299
Digital Object Identifier: 10.1214/15-AOP1087

Primary: 60F05 , 60K35

Keywords: generalized Rosenblatt processes , long memory , Rosenblatt processes , Self-similar processes

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 2 • March 2017
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