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.
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