Older references are still quite useful. Chung's A course in probability theory used to be the standard text for courses on convergence of random variables. Lamperti's Probability is a short and conversational book, in much the same spirit as Rosenthal. Breiman's Probability is a readable but comprehensive treatment, in the same spirit as Durrett. And there are parts of the theory that still are best learned from an even older and more encyclopaedic text, namely Feller's An introduction to probability theory and its applications, both volume 1 and volume 2.