Summary: An important model in communications is the stochastic FM signal \[ s_t= A\cos\left(t\omega_c+ \sum^t_{k= 1}m_k+ \theta_0\right), \] where the message process \(\{m_t\}\) is a stochastic process. We investigate the linear models and limit distributions of FM signals. Firstly, we show that this nonlinear model in the frequency domain can be converted to an \(\text{ARMA}(2,q+1)\) model in the time domain when \(\{m_t\}\) is a Gaussian \(\text{MA}(q)\) sequence. The spectral density of \(\{s_t\}\) can then be solved easily for MA message processes. Also, an error bound is given for an ARMA approximation for more general message processes. Secondly, we show that \(\{s_t\}\) is asymptotically strictly stationary if \(\{m_t\}\) is a Markov chain satisfying a certain condition on its transition kernel. Also, we find the limit distribution of \(s_t\) for some message processes \(\{m_t\}\). These results show that a joint method of probability theory, linear and nonlinear time series analysis can yield fruitful results. They also have significance for FM modulation and demodulation in communications.