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We theoretically and numerically investigate the spin fluctuations induced in a thermal atomic ensemble by an external fluctuating uniaxial magnetic field, in the context of a standard spin noise spectroscopy (SNS) experiment. We show that additional spin noise is excited, which dramatically depends on the magnetic noise variance and bandwidth, as well as on the power of the probe light and its polarization direction. We develop an analytical perturbative model proving that this spin noise first emerges from the residual optical pumping in the medium, which is then converted into spin fluctuations by the magnetic noise and eventually detected using SNS. The system studied is a spin-1 system, which thus shows both Faraday rotation and ellipticity noises induced by the random magnetic fluctuations. The analytical model gives results in perfect agreement with the numerical simulations, with potential applications in future experimental characterization of stray field properties and their influence on spin dynamics., Comment: 16 pages, 10 figures, submitted to New Journal of Physics