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Cosmological linear perturbation theory predicts that the peculiar velocity$V(x)$ and the matter overdensity $\delta(x)$ at a same point $x$ arestatistically independent quantities, as log as the initial densityfluctuations are random Gaussian distributed. However nonlinear gravitationaleffects might change the situation. Using framework of second-orderperturbation theory and the Edgeworth expansion method, we study local densitydependence of bulk velocity dispersion that is coarse-grained at a weaklynonlinear scale. For a typical CDM model, the first nonlinear correction ofthis constrained bulk velocity dispersion amounts to $\sim 0.3\delta$ (Gaussiansmoothing) at a weakly nonlinear scale with a very weak dependence oncosmological parameters. We also compare our analytical prediction withpublished numerical results given at nonlinear regimes.Comment: 16 pages including 2 figures, ApJ 537 in press (July 1

Nonlinear Velocity‐Density Coupling: Analysis by Second‐Order Perturbation Theory