Using the method of generative processes, we construct a model of a random linear symmetric stable Markov process, which is a natural generalization of the Markov Gaussian process and preserves its main property: invariance with respect to arbitrary linear transformations. Methods for analyzing such processes are developed. In particular, it is proposed to use the information correlation function as a characteristic of the pair dependence between the process values at different times. This function is used to calculate the determination interval of the process.