Motivated by stochastic models of climate phenomena, the steady state of a linear stochastic model with additive Gaussian white noise is studied. Fluctuation theorems for nonequilibrium steady states provide a constraint on the character of these fluctuations. The properties of the fluctuations which are unconstrained by the fluctuation theorem are investigated and related to the model parameters. The irreversibility of trajectory segments, which satisfies a fluctuation theorem, is used as a measure of nonequilibrium fluctuations. The moments of the irreversibility probability density function (pdf) are found and the pdf is seen to be non-Gaussian. The average irreversibility goes to zero for short and long trajectory segments and has a maximum for some finite segment length, which defines a characteristic time scale of the fluctuations. The initial average irreversibility growth rate is equal to the average entropy production and is related to noise amplification. For systems with a separation of deterministic time scales, modes with time scales much shorter than the trajectory time span and whose noise amplitudes are not asymptotically large, do not, to first order, contribute to the irreversibility statistics, providing a potential basis for dimensional reduction.