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

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 orders and 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.M. Clausel's research was partially supported by the PEPS project AGREE and LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d' avenir. Francois Roueff's research was partially supported by the ANR project MATAIM NTO9 441552. Murad S. Taqqu was supported in part by the NSF grants DMS-1007616 and DMS1309009 at Boston University. Ciprian Tudor's research was supported by the CNCS grant PNII-ID-PCCE-2011-2-0015 (Romania). (ANR-11-LABX-0025-01 - PEPS project AGREE and LabEx PERSYVAL-Lab; French program Investissement d' avenir; 441552 - ANR project MATAIM NTO9; DMS-1007616 - NSF; DMS1309009 - NSF; PNII-ID-PCCE-2011-2-0015 - CNCS (Romania))Accepted manuscrip