Improving the resolution of fluorescence microscopy beyond the diffraction limit can be achieved by acquiring and processing multiple images of the sample under different illumination conditions. One of the simplest techniques, Random Illumination Microscopy (RIM), forms the super-resolved image from the variance of images obtained with random speckled illuminations. However, the validity of this process has not been fully theorized. In this work, we characterize mathematically the sample information contained in the variance of diffraction-limited speckled images as a function of the statistical properties of the illuminations. We show that an unambiguous two-fold resolution gain is obtained when the speckle correlation length coincides with the width of the observation point spread function. Last, we analyze the difference between the variance-based techniques using random speckled illuminations (as in RIM) and those obtained using random fluorophore activation (as in Super-resolution Optical Fluctuation Imaging, SOFI).