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This paper is concerned with the problems of stability and stabilization for discrete‐time switched linear systems with a new class of switching signals. The considered switching signals are of random mode‐dependent sojourn‐time (RMST) property, which is capable of describing the mode‐dependent sojourn‐time period consisting of a fixed part with no switching and a random part with known sojourn‐time expectation. Compared with the previous studies assuming that the switching dynamics of the random part can be described by a Markov chain, the length of the random part is allowed to obey any distribution including geometric distribution. Besides, the mode switching of RMST switching is arbitrary indicating that it requires no statistical information of transition probabilities. The criteria of stability and stabilization are derived for the RMST switched systems. To further reduce the limitation of RMST switching on switching frequency, a concept of random mode‐dependent persistent sojourn‐time (RMPST) is proposed. Then, in the presence of asynchronous switching, the conditions of the existence of controllers are presented for the switched systems with persistent sojourn‐time expectation. A numerical example and a practical example of a space robot manipulator are given to demonstrate the effectiveness and applicability of the developed theoretical results.