This paper investigates an H-infinity filtering problem for discrete-time systems with randomly varying sensor delays. The stochastic variable involved is a Bernoulli distributed white sequence appearing in measured outputs. This measurement mode can be used to characterize the effect of communication delays and/or data-loss in information transmissions across limited bandwidth communication channels over a wide area. H-infinity filtering of this class of systems is used to design a filter using the measurements with random delays to ensure the mean-square stochastic stability of the filtering error system and to guarantee a prescribed H-infinity filtering performance. A sufficient condition for the existence of such a filter is presented in terms of the feasibility of a linear matrix inequality (LMI). Finally, a numerical example is given to illustrate the effectiveness of the proposed approach. (C) 2008 Elsevier Ltd. All rights reserved.