Convergence of types under monotonous mappings

Let $\mathcal F$ be the set of all D.F. on $\overline{\mathbf R}{}^d=[-\infty,\infty)^d$. Denote by $GMA$ the group of all max-automorphisms of $\overline{\mathbf R}{}^d$, i.e. such one-to-one mappings $L$ that preserve the max-operation in $\overline{\mathbf R}{}^d$, $L(x\vee y)=L(x)\vee L(y)$. We define type $(F):=\{G\in\mathscr{F}:\exists T\in GMA,G=F\circ T\}$. Hеге the convergence to type theorem is proved for distributions in $\mathcal F$ and norming sequences $\{L_n\}$ in $GMA$.