[en] The long-time behavior of a system is suggested to confirm nonergodicity of non-Markovian Brownian dynamics, namely, whether the stationary probability density function (PDF) of the system characterized mainly by low moments of variables depends on the initial preparation. Thus we classify nonergodic Brownian motion into two classes: the class-I is that the PDF of a force-free particle depends on the initial velocity and the equilibration can be recovered through a bounded potential; while the PDF in the class-II depends on the initial coordinate and the equilibration can not be approached by introducing any potential. We also compare our result with the conditions of three kinds for ergodicity. (general)