Limit theorems for compositions of distributions on some nilpotent Lie groups

Let $X_1,X_2,\dots$ be a sequence of independent random elements of unipotent group $G$ (upper triangular matrices with 1's on the diagonal) with the same distribution $\mu$ on $G$. Asymptotical behaviour of the distribution $\mu^n$ of the product $X(n)=X_1X_2\dots X_n$ is studied.
It is shown that the distribution of the properly normalized product $X(n)$ weakly converges to the distribution of $Z(1)$, where $Z(t)$ is an invariant Brownian motion on some nilpotent Lie group $G_\mu$ with the same space as that of $G$ and the multiplication rule dependent on the measure $\mu$. Necessary and sufficient conditions are obtained for the two compositions $\mu_1^n$ and $\mu_2^n$ to come together as $n\to\infty$.