## 1 Introduction  
Stochastic dynamical systems widely exist in scientific research and engineering applications. When systems or engineering structures are subjected to random excitations, such as earthquakes, winds, waves, and so on, they will show obviously random behaviors and response states. Random responses may cause very serious consequences, such as the collapse of engineering structures under the action of earthquakes, the vibration of high-rise building structures under the action of random wind fields, and the damage of sea-facing structures under the action of catastrophic waves. Therefore, in engineering applications, it is necessary to analyze the response of structures in the time domain. For the spatial structure of a complex system, it is necessary to observe the response of the structure under the action of random inputs to verify its reliability. The random inputs are generally converted from the generated time history records. The creation of reasonable progress describing the stochastic process has great research significance. Existing research on stochastic processes is mainly divided into two categories. One is the physical modeling method, which determines the expression of the random process after considering the physical mode of random excitation. In this method, the distribution of random variables can be obtained by analyzing the measured field data. However, the number of data obtained from real measurements are generally very small, so it is impossible to rigorously analyze the random process. Therefore, in the artificial simulation of random processes, another kind of method is then adopted, that is, the numerical simulation. This method can well generate samples of random processes by combining the probability information of random processes, such as finite-dimensional joint distribution, the joint moment between finite time, power spectral density (PSD) function, and so on. Based on mathematical analysis, many scholars have developed different methods including the Karhunen-Loeve decomposition method [1; 2], sampling theorem method [3], SRM [4; 5; 6], wavelet transform [7; 8], and so on. Among them, the SRM has been widely used for its convenience and robustness. The SRM originated from [PERSON]'s analysis of one-dimensional univariate random noise [9]. In 1972, [PERSON] [4] proposed the SRM and used it to simulate multi-dimensional and multi-variable stationary random processes. Later, [PERSON] and [PERSON] [6; 10] proved that the simulation results of this method were Gaussian, unbiased, and ergodic. Since then, many scholars have extended this method to simulate the wind speed field [11; 12], seismic ground motion [13; 14], and ocean wave motion [15; 16], among others. In addition, it is worth mentioning that [PERSON] [10] proposed a method to generate a quasi-stationary spectrum compatible ground motion vector process in 1996, in which fast Fourier transform (FFT) is combined to improve the calculation efficiency of SRM, and this improvement makes SRM a classic method for random process simulation. On the basis of SRM and combined with [PERSON]'s evolutionary spectrum theory [17; 18], [PERSON] et al. [19] derived a spectral representation formula for simulating non-stationary random processes. In recent decades, the wavelet method has been widely used in various fields because of its good time-frequency resolution characteristics, flexible selection of wavelet bases, and higher computational efficiency. On the one hand, the non-stationary PSD can be estimated via the wavelet method. [PERSON] and [PERSON] [20; 21] established the relationship between the mean square value of the wavelet transform coefficients and the PSD. [PERSON] and [PERSON] [22] derived a method for estimating the EPSD using wavelet transform. [PERSON] and [PERSON] [23] extended the wavelet method to simulate a multi-dimensional stochastic process and estimate the EPSD of strong wind. On the other hand, many scholars use wavelet to simulate random processes.