The angular correlation function of the cosmic microwave background temperature fluctuations observed by the COBE DMR experiment sets constraints on the shape and amplitude of the primordial density fluctuation spectrum. We discuss these constraints in the light of detailed Monte Carlo simulations. We find that a large range of primordial spectral indices n can still be allowed from present data, although n = 1 is greatly favored. For the scale-invariant case we find an estimate for the ensemble-average quadrupole which has almost half of the uncertainty and is slightly smaller than the previously quoted one. We also consider the effects of a possible background of gravitational waves in CDM tilted models and derive from the DMR data the relevant values for the biasing parameter.