Affine stochastic equation with triangular matrices

TL;DR

This paper analyzes the tail behavior of solutions to a stochastic affine equation with upper triangular matrices, deriving precise asymptotics for the distribution tails of the solution components.

Contribution

It provides sharp asymptotic results for the tail distributions of solutions to stochastic equations with triangular matrices, extending Kesten-Goldie theory.

Findings

Tail of X_2 follows a power law t^{-a}.

Tail of X_1 follows t^{-a} with a logarithmic correction.

Results generalize existing tail asymptotics for stochastic equations.

Abstract

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.