We perform a linear stability analysis of a dust layer in a turbulent gas disk. Youdin investigated the secular gravitational instability (GI) of a dust layer using hydrodynamic equations with a turbulent diffusion term. We obtain essentially the same result independently of Youdin. In the present analysis, we restrict the area of interest to small dust particles, while investigating the secular GI in a more rigorous manner. We discuss the time evolution of the dust surface density distribution using a stochastic model and derive the advection-diffusion equation. The validity of the analysis by Youdin is confirmed in the strong drag limit. We demonstrate quantitatively that the finite thickness of a dust layer weakens the secular GI and that the density-dependent diffusion coefficient changes the growth rate. We apply the results obtained to the turbulence driven by the shear instability and find that the secular GI is faster than the radial drift when the gas density is three times as large as that in the minimum-mass disk model. If the dust particles are larger than chondrules, the secular GI grows within the lifetime of a protoplanetary disk.
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