In this paper, we propose self-supervised speaker representation learning strategies, which comprise of a bootstrap equilibrium speaker representation learning in the front-end and an uncertaintyaware probabilistic speaker embedding training in the back-end. In the front-end stage, we learn the speaker representations via the bootstrap training scheme with the uniformity regularization term. In the backend stage, the probabilistic speaker embeddings are estimated by maximizing the mutual likelihood score between the speech samples belonging to the same speaker, which provide not only speaker representations but also data uncertainty. Experimental results show that the proposed bootstrap equilibrium training strategy can effectively help learn the speaker representations and outperforms the conventional methods based on contrastive learning. Also, we demonstrate that the integrated two-stage framework further improves the speaker verification performance on the VoxCeleb1 test set in terms of EER and MinDCF.INDEX TERMS Speaker verification, self-supervised learning, bootstrap representation learning, probabilistic speaker embedding.