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We use techniques of Malliavin calculus to study the convergence in law of a family of generalized Hermite processes Zγ with kernels defined by parameters γ taking values in a tetrahedral region ∆ of R q . We prove that, as γ converges to a face of ∆, the process Zγ converges to a compound Gaussian distribution with random variance given by the square of a Hermite process of one lower rank. The convergence in law is shown to be stable. This work generalizes a previous result of Bai and Taqqu,doi:10.1214/17-ecp99 fatcat:swyp26swq5fozhkc5cwiashe5y