Limit theorems for products of independent triangular matrices

The aim of the present paper is to study the limit distribution for the complete group of triangular matrices with non-negative elements on the diagonal.
It is shown, that the distribution of the properly normalized product $G_n$ converges weakly to the distribution of $W^l$, where $W^l$ is the triangular matrix elements of which are some functionals of an $l$-dimensional Wiener process.
An explicit form of the probability density is obtained in the case of random matrices $2\times 2$. The probability density of the maximum of some stationary process is also obtained.