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 patterns (periodic grids, focused beams, or more generally speckles). When the illuminations are known, the super-resolved reconstruction is generally formed from a linear combination of the multiple diffraction-limited images, and the resolution gain is easily determined. On the other hand, when the illuminations are unknown, the resolution gain is seldom well-defined. In this work, we consider the recent Random Illumination Microscopy (RIM) technique where the illuminations are unknown speckles and the reconstructions are formed from the variance of the images. We show that an unambiguous two-fold resolution gain can be obtained only 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).