Spontaneous fluctuations are a robust phenomenon of plasmas near equilibrium, which does not depend on the existence of instabilities, and their characterization provides valuable information about the role of the relativistic temperatures T similar to mc(2) in relativistic regimes. In this paper, we study the broadband spontaneous electromagnetic fluctuations in a relativistic and magnetized pair plasma described by Maxwell-Juttner distribution functions, based on the fluctuation-dissipation theorem and particle-in-cell simulations. It is shown that the transverse electromagnetic fluctuations are constrained by a temperature-independent critical frequency omega(c) = root Omega(2)(c) + (ck)(2), which separates the collisionless damping regime (vertical bar omega vertical bar < omega(c)) from the case where the waves can propagate without damping (vertical bar omega vertical bar > omega(c)). Further, relativistic simulations show an excellent agreement between analytical and numerical results. We expect this formalism to contribute to obtaining useful physical information about the state of relativistic plasmas.