This letter proposes new estimations of fractional power spectral density (FrPSD) and fractional correlation function (FrCF) for nonuniform sampling of random signals with non-stationarity and limited bandwidths in the fractional Fourier domain. Unlike previous works, the developed FrPSD and FrCF estimations are capable of dealing with unknown sampling instants. In order to obtain them, we first formulate approximations of FrCF and FrPSD making use of uniform sampling instants. Then we convert the approximate FrPSD to a fractional filtered version of the FrPSD for the original random signal, which does not rely on the sampling instants. With such operations, we propose the FrPSD estimation to cancel the bias of FrPSD approximation by means of a fractional inverse filtering and thereby obtain a high accuracy of it. The FrCF estimation is proposed to be the inverse fractional Fourier transform of the FrPSD, and it serves as the fractional interpolation of the previously obtained approximation of the FrCF. Simulation results show the effectiveness of the proposed estimation methods.