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We have measured on picosecond time scales the polarization behavior of vertical-cavity surface-emitting lasers (VCSELs) pumped with short optical pulses. Our data describe the temporal properties of the intensity and intensity noise for orthogonally polarized modes as well as noise correlations between the modes in both the circular and the linear detection bases. A variety of behavior is observed that depends on the polarization of the pump, the strength of the pump, the VCSEL operating temperature which determines the carrier spin-flip rates, and the degree of birefringence and dichroism in the laser. We develop an extension of the "split density" model which includes carrier relaxation processes that are important on picosecond time scales. Our numerical simulations illustrate how spin-flip processes lead to anti-correlated fluctuations in the circularly polarized modes and how anti-correlated noise can arise between the linearly polarized modes without switching between two "preferred" modes, as is commonly seen on longer time scales.