In this paper we define a kind of vanishing Carleson measure on $$\mathbb {R}^{n+1}_+$$ R + n + 1 and give its characterization by the compact property of some convolution operator. We also investigate the construction of vanishing Carleson measures generated by a family of the multilinear operators $$\{\Theta _t\}_{t>0}$$ { Θ t } t > 0 and $$CMO$$ C M O functions. As some applications of our results, we also give the boundedness and compactness for the paraproduct $$\pi _{\vec {b}}$$ π b → associated with the family $$\{\Theta _t\}_{t>0}$$ { Θ t } t > 0 on $$L^2(\mathbb {R}^n)$$ L 2 ( R n ) , which is defined by $$\begin{aligned} \pi _{\vec {b}}(f)(x)= \int _0^\infty \eta _t*\big ((\varphi _t*f)\Theta _t(b_1,\ldots ,b_m)\big )(x)\; \frac{dt}{t}. \end{aligned}$$ π b → ( f ) ( x ) = ∫ 0 ∞ η t ∗ ( ( φ t ∗ f ) Θ t ( b 1 , … , b m ) ) ( x ) d t t . Further, for the linear case (i.e., $$m=1$$ m = 1 ), we show that the paraproduct $$\begin{aligned} B_b(f)(x)=\int _0^\infty (f*\varphi _t)(x)(b*\psi _t)(x)\frac{\alpha (t)}{t}dt, \end{aligned}$$ B b ( f ) ( x ) = ∫ 0 ∞ ( f ∗ φ t ) ( x ) ( b ∗ ψ t ) ( x ) α ( t ) t d t , which was introduced by Coifman and Meyer, is also a compact operator on $$L^2(\mathbb {R}^n)$$ L 2 ( R n ) if $$b\in CMO(\mathbb {R}^n)$$ b ∈ C M O ( R n ) and $$\alpha \in L^\infty (\mathbb {R}^n)$$ α ∈ L ∞ ( R n ) .