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We study the d-dimensional branching random walk for d ≥ 3. This process has extremal equilibria for every intensity. We are interested in the large space scale and large space-time scale behavior of the equilibrium state. We show that the fluctuations of space and space-time averages with a non-classical scaling are Gaussian in the limit. For this purpose we use the historical process, which allows a family decomposition. To control the distribution of the families we use the concept ofdoi:10.1214/ejp.v7-106 fatcat:w6kjsb7qwzeuvmqli7etlicgyu