The use of a moving, random aperture in the plane of a hologram has been recently suggested for reducing speckle in the reconstructed image of a diffusing object. A theoretical analysis shows that while the attenuation of the power spectrum of the speckle is proportional to the square of the average transmittance of the random aperture, the power spectrum of the signal is also attenuated by the same factor, resulting in a loss of image contrast. Because of this there is no overall gain. Some experimental results are presented which support these conclusions.