Asymptotic behavior of the quadratic variation of the sum of two Hermite processes of consecutive orders

Abstract

Hermite processes are self--similar processes with stationary increments which appear as limits of normalized sums of random variables with long range dependence. The Hermite process of order $1$ is fractional Brownian motion and the Hermite process of order $2$ is the Rosenblatt process. We consider here the sum of two Hermite processes of order $q\ge 1$ and $q+1$ and of different Hurst parameters. We then study its quadratic variations at different scales. This is akin to a wavelet decomposition. We study both the cases where the Hermite processes are dependent and where they are independent. In the dependent case, we show that the quadratic variation, suitably normalized, converges either to a normal or to a Rosenblatt distribution, whatever the order of the original Hermite processes.

References in corpus (2)

• Properties and numerical evaluation of the Rosenblatt distribution
• Large scale behavior of wavelet coefficients of non-linear subordinated processes with long memory

Cited by in corpus (3)

• Limiting behavior of large correlated Wishart matrices with chaotic entries
• Wavelet eigenvalue regression for $n$-variate operator fractional Brownian motion
• Non-Central Limit Theorem for Quadratic Functionals of Hermite-Driven Long Memory Moving Average Processes