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This paper is devoted to the research of the isomorphic multidimensional cyclic structure and multidimensional phase structure of the cyclic random process (CRP) and to its formation method, which enables a rigorous formalization of intuitive ideas concerning cyclic stochastic motion. The fundamental properties of the cyclic random process and analytical dependencies between the multidimensional cyclic structure, multidimensional phase structure and rhythm structure of the CRP have been established. This work shows that the CRP is able to take into account the cyclicity of multidimensional distribution functions of cyclic signals as well as the variability in the rhythm of the investigated signals. A subclass of the CRP is the periodic random process, which allows for the use of classical processing methods of cyclic signals with a regular rhythm. Based on a series of experiments, significant advantages of the CRP as a mathematical model of electrocardiographic signals (ECG) compared to the periodic random process are shown.