### Limit currents and value distribution of holomorphic maps

We construct $d$-closed and $d{d}^{c}$-closed positive currents associated to a holomorphic map $\phi $ via cluster points of normalized weighted truncated image currents. They are constructed using analogues of the Ahlfors length-area inequality in higher dimensions. Such classes of currents are also referred to as Ahlfors currents. We give some applications to equidistribution problems in value distribution theory.