  We study bivariate stochastic recurrence equations with triangular matrix
coefficients and we characterize the tail behavior of their stationary
solutions ${\bf W} =(W_1,W_2)$. Recently it has been observed that $W_1,W_2$
may exhibit regularly varying tails with different indices, which is in
contrast to well-known Kesten-type results. However, only partial results have
been derived. Under typical "Kesten-Goldie" and "Grey" conditions, we
completely characterize tail behavior of $W_1,W_2$. The tail asymptotics we
obtain has not been observed in previous settings of stochastic recurrence
equations.
