On the rate of convergence of linear combinations of absolute order statistics to the normal law

Let $\{X_j\}$ ($j=1,2,\dots,n$) be a sequence of symmetric independent identically distributed random variables and $\{X_{j,n}\}$ ($j=1,2,\dots,n$) be the corresponding absolute order statistics, i.e. $|X_{1,n}|\le|X_{2,n}|\le\dots\le|X_{n,n}|$.
Some results are obtained for the rate of convergence of linear combinations of the random variables $X_{j,n}$ to the normal law.