  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
