The implementation of fluctuations in the lattice Boltzmann method has made significant progress in the last 10 years. The significance of incorporating noise to all non-conserved degrees of freedom was a significant recent discovery that was based on a simplified Langevin treatment of the linarized Boltzmann equation. However, for non-vanishing mean velocities significant deviations in the correlation functions were observed. In this thesis we show how we can largely alleviate these deviations by incorporating fully velocity dependent moment transforms and thus recover a fluctuation dissipation theorem that is valid for a larger range of velocities. Furthermore we show that the remaining deviations can be attributed to the collision operator of the linearized Boltzmann equation not being identical to the one of the BGK collision which forms the basis of most modern lattice Boltzmann applications. Finally we show that the locally velocity dependent transforms significantly improve the stability of fluctuating lattice Boltzmann simulations at low particle densities.