The Annals of Probability

Behavior of the generalized Rosenblatt process at extreme critical exponent values

Shuyang Bai and Murad S. Taqqu

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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 $L^{2}(\Omega)$.

Article information

Ann. Probab., Volume 45, Number 2 (2017), 1278-1324.

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

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Long memory self-similar processes Rosenblatt processes generalized Rosenblatt processes


Bai, Shuyang; Taqqu, Murad S. Behavior of the generalized Rosenblatt process at extreme critical exponent values. Ann. Probab. 45 (2017), no. 2, 1278--1324. doi:10.1214/15-AOP1087.

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