In this paper we consider Gaussian RBF kernels support vector machine classification (SVMC) algorithm with uniformly ergodic Markov chain (u.e.M.c.) samples in reproducing kernel Hilbert spaces (RKHS). We analyze the learning rates of Gaussian RBF kernels SVMC based on u.e.M.c. samples and obtain the fast learning rate of Gaussian RBF kernels SVMC based on u.e.M.c. samples by using the strongly mixing property of u.e.M.c. samples. We also present the numerical studies on the learning performance of Gaussian RBF kernels SVMC based on Markov sampling for real-world datasets. These experimental results show that Gaussian RBF kernels SVMC based on Markov sampling has better learning performance compared to randomly independent sampling.