In this paper, we propose a new approach for sequence classification which is based on the framework of Reproducing Kernel Hilbert Spaces (RKHS). We introduce the theoretical material which leads to the formulation of an original sequence kernel, that we implement in a SVM scheme. Experiments are carried out on a speaker verification task using NIST SRE data. They show that our new sequence kernel significantly outperforms the generative approach with Gaussian Mixture Models (GMM). They also show that it generally outperforms the powerful Generalized Linear Discriminant Sequence (GLDS) kernel, while offering more efficiency and flexibility (than GLDS).