Behavior of the generalized Rosenblatt process at extreme critical exponent values

TL;DR

This paper investigates the behavior of the generalized Rosenblatt process as its defining exponents approach boundary values, revealing non-Gaussian limits on two boundaries and Brownian motion on the third, with detailed convergence rates.

Contribution

It characterizes the limiting behavior of the generalized Rosenblatt process at boundary critical exponents, including convergence rates and dependence on approach directions.

Findings

Limits are non-Gaussian on two boundaries.

Limit is Brownian motion on the third boundary.

Convergence rates to boundary limits are established.

Abstract

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 $L^2(\Omega)$.