  We report a detailed investigation on the properties of correlation spectra
for cold atoms under the condition of Electromagnetically Induced Transparency
(EIT). We describe the transition in the system from correlation to
anti-correlation as the intensity of the fields increases. Such transition
occurs for laser frequencies around the EIT resonance, which is characterized
by a correlation peak. The transition point between correlation and
anti-correlation is independent of power broadening and provides directly the
ground-state coherence time. We introduce a method to extract in real time the
correlation spectra of the system. The experiments were done in two distinct
magneto-optical traps (MOT), one for cesium and the other for rubidium atoms,
employing different detection schemes. A simplified theory is introduced
assuming three-level atoms in $\Lambda$ configuration interacting with a laser
with stochastic phase fluctuations, providing a good agreement with the
experimental observations.
