This article addresses the problem of distributed state estimation on the infinite-time horizon for discrete-time systems with multiple sensors. The desired filter, referred to as distributed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> Gaussian filter, reaching a tradeoff between optimality and robustness, constructs a local estimate based on its own observation and on those collected from its neighbors. A Nash game approach is used to deal with such a multiobjective distributed filtering problem, and causal bounded Nash equilibrium strategies, consisting of the optimal filter gains and the corresponding worst-case disturbance signals, are analytically conducted based on solutions to a set of cross-coupled algebraic Riccati equations. Moreover, to further improve the cohesiveness among local estimates, additional consensus objective is taken into consideration by exchanging prior estimates among neighboring nodes. A choice of the consensus term, with which the error dynamics are shown to be internally stable, is derived via a numerically tractable and efficient method. Finally, an application to speed monitoring through wireless sensor network is included to show the validity of the presented results.