The complex Banach spaces X with values in which every bounded holomorphic function in the unit ball B of Cd(d > 1) has boundary limits almost surely are exactly the spaces with the analytic Radon-Nikodym property.The proof is based on inner Hardy martingales introduced here.The inner Hardy martingales are constructed in terms of inner functions in B and are reasonable discrete approximations for the image processes of the holomorphic Brownian motion under X-valued holomorphic functions in B.