The linear canonical transform (LCT) plays an important role in optical and digital signal processing. Over the past few decades, there has been a vast amount of research on sampling theorems for a deterministic signal bandlimited in the LCT domain. However, signals are usually random in practical situations. Hence in this paper, we study sampling theorems for a random signal bandlimited in the LCT domain. We first construct a random signal theoretic framework in the LCT domain, such as the LCT power spectral density and the LCT auto-correction function. Then, we formulate uniform sampling theorem and multi-channel sampling theorem for a random signal bandlimited in the LCT domain. Finally, we analyze two kinds of reconstruction error estimates for uniformly sampling a random signal in the LCT domain: aliasing error and truncation error.