Fluctuations of the white-light supercontinuum produced by ultrashort laser pulses in self-guided filaments (spatiotemporal solitons) in air are investigated. We demonstrate that correlations exist within the white-light supercontinuum, and that they can be used to significantly reduce the laser intensity noise by filtering the spectrum. More precisely, the fundamental wavelength is anticorrelated with the wings of the continuum, while conjugated wavelength pairs on both sides of the continuum are strongly correlated. Spectral filtering of the continuum reduces the laser intensity noise by 1.2dB, showing that fluctuations are rejected to the edges of the spectrum.