We consider the distributed detection problem of a temporally correlated random radio source signal using a wireless sensor network capable of measuring the energy of the received signals. It is well-known that optimal tests in the Neyman-Pearson setting are based on likelihood ratio tests (LRT), which, in this set-up, evaluate the quotient between the probability density functions (PDF) of the measurements when the source signal is present and absent. When the source is present, the computation of the joint PDF of the energy measurements at the nodes is a challenging problem. This is due to the statistical dependence introduced to the received signals by the radio source propagated through fading channels. We deal with this problem using the characteristic function of the (intractable) joint PDF, and proposing an approximation to it. We derive bounds for the approximation error in two wireless propagation scenarios, slow and fast fading, and show that the proposed approximation is exponentially tight with the number of nodes when the time-bandwidth product is sufficiently high. The approximation is used as a substitute of the exact joint PDF for building an approximate LRT, which performs better than other well-known detectors, as verified by Monte Carlo simulations.