In this paper, we consider the distributed sensor fusion problem over sensor networks under directed communication links and bandwidth constraint. We investigate the impact of the additive quantization model on the proposed two-stage averaging based algorithm. Existing works on the effect of the additive model show that convergence can be guaranteed only if the quantization error variances form a convergent series. We show that the proposed algorithm achieves the performance of the optimal centralized estimate even if the quantization error variances are not vanishing. This is guaranteed by establishing a law of the iterated logarithm for weighted sums of independent random vectors. Moreover, an explicit bound of the convergence rate of the proposed algorithm is given to quantify its almost sure performance. (C) 2017 Elsevier Ltd. All rights reserved.