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Affine stochastic equation with triangular matrices
Ewa Damek, Jacek Zienkiewicz · 2018-06-23 · via math.PR updates on arXiv.org

We study solution X of the stochastic equation X = AX +B, where A is a random matrix and B,X are random vectors, the law of (A,B) is given and X is independent of (A,B). The equation is meant in law, the matrix A is 2x2 upper triangular, A_{11}=A_{22}>0, A_{12} is real. A sharp asymptotics of the tail of X =(X _1,X_2) is obtained. We show that under "so called" Kesten-Goldie conditions P (X_2>t)\sim t^{-a} and P (X_1>t )\sim t^{-a}(\log t)^b, where b =a or a\2.

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