Particle picture representation of the non-symmetric Rosenblatt process and Hermite processes of any order

We provide a particle picture representation for the non-symmetric Rosenblatt process and for Hermite processes of any order, extending the result of Bojdecki, Gorostiza and Talarczyk in~\cite{FILT}. We show that these processes can be obtained as limits in the sense of finite-dimensional distributions of certain functionals of a system of particles evolving according to symmetric stable L\'{e}vy motions. In the case of $k$-Hermite processes the corresponding functional involves $k$-intersection local time of symmetric stable L\'{e}vy processes