Central limit theorem for weakly dependent random variables with values in $D\left[0,1\right]$

The main goal of this article is to prove the central limit theorem for sequences of random variables with values in the space $D\left [0,1\right]$. We assume that the sequence satisfies the mixing conditions. In the paper the central limit theorems for sequences with strong mixing and $\rho_{m}$-mixing are proved.