Boundary Behavior of Nonlocally Convex Vector-Valued Hardy Classes in the Complex Ball

For a complex quasi-Banach space X, every bounded X-valued holomorphic function in the unit ball B of C^d (d > 1) has radial limits almost surely if and only if all bounded plurisubharmonic martinga les in X converge almost surely.The proof is based on inner Hardy martingales. The inner Hardy martingales are constructed in terms of inner functions in B and are reasonable discrete approximations for the image processes of Brownian motion in B under X-valued holomorphic functions in B.