In this paper, we focus on the second-order stochastic processes and the ergodicity and stationarity properties of their wavelet transforms (WT), and concern about both of the continuous-time and the discrete-time cases. The aim of the paper is to show that the ergodicity property of a second-order stochastic process is preserved by WT. Moreover, under some soft constraints for wavelet functions, the WT of a second-order process with wide-sense stationary increments/jumps or wide-sense stationary (W.S.S.) property is W.S.S. and ergodic if the process is ergodic. The fractional Brownian motion (fBm) processes have been used in many research areas of 1/f-type noises, fractals, image textures, etc.. But, these researches did not deal with the calculation problems of the fBm processes in practice. Actually, the ergodicity property of the fBm process is not concluded by the ergodicity theorem. In our work, the ergodicity property of the WT of an fBm process would be certified too.