A non-iterative algorithm for simulating non-Gaussian random process was proposed to avoid the divergence problem in iteration. Firstly, based on the nonlinear translation process, the conversion relationship between latent Gaussian stochastic process and non-Gaussian one was analyzed in detail. Then, it was proved with the reduction to absurdity that the target power spectral density (PSD) and the marginal probability distribution (MPD) function for an arbitrary non-Gaussian random process being compatible is necessary. Thereby, the criterion to judge the compatibleness between the target PSD and the MPD was established (i.e., the target PSD function of the latent Gaussian stochastic process must be a nonnegative function). Furthermore, the modification programs were developed for the case of the target PSD and the MPD being incompatible, a non-iterative algorithm was proposed for simulating a non-Gaussian random process with a single variable. Finally, a non-Gaussian fluctuating wind pressure with different skewness was simulated with the proposed non-iterative algorithm. Its feasibility and validity were verified through comparing correlation functions, PSD, and MPD with their corresponding targets.