Affine stochastic equation with triangular matrices

Abstract

We study solution $\mathcal{X}$ of the stochastic equation $$\mathcal{X}=A\mathcal{X}+B,$$

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 $2\times 2$ upper triangular, $A_{11}=A_{22}>0$, $A_{12}\in \mathbb{R}$. A sharp asymptotics of the tail of $\mathcal{X}=({\mathcal{X}}_1,{\mathcal{X}}_2)$ is obtained. We show that under ‘so called’ Kesten–Goldie conditions $\mathbb{P}({\mathcal{X}}_2>t)\sim t^{-\alpha }$ and $\mathbb{P}({\mathcal{X}}_1>t)\sim t^{-\alpha }{(logt)}^{\tilde{\alpha }}$, where $\tilde{\alpha }=\alpha$ or $\alpha /2$.

Keywords:

• Matrix recursion
• multivariate affine stochastic equation
• regular behavior at infinity
• stationary solution
• triangular matrices

AMS Subject Classifications:

• Primary: 60G10
• 60J05
• 62M10
• Secondary: 60B20
• 91B84

Notes

No potential conflict of interest was reported by the authors.

1 The statement in [10] is much more general than what we need here and the proof is quite advanced. If there is $\epsilon >0$ such that $\mathbb{E}{a}^{\epsilon }<1$ and $\mathbb{E}(|y{|}^{\epsilon }+|b_1{|}^{\epsilon }+|b_2{|}^{\epsilon })<\infty$, then negativity of the Lapunov exponent follows quite easily, see [23], Proposition 7.4.5 and e.g [9]. Finiteness of the above moments is assumed here anyway, see (2.6(2.6) $$\begin{array}{cc}\hfill \mathbb{E}{a}^{\alpha }=1\text{and}0& <\rho =\mathbb{E}{a}^{\alpha }loga<\infty ,\hfill \end{array}$$(2.6) ), (2.7(2.7) $$\begin{array}{cc}\hfill \mathbb{E}(|b_1{|}^{\alpha }+|b_2{|}^{\alpha })& <\infty \hfill \end{array}$$(2.7) ) and (2.12(2.12) $$\begin{array}{c}\hfill \mathbb{E}{a}^{\alpha +{\epsilon }_0}<\infty ,\mathbb{E}|b_2{|}^{\alpha +{\epsilon }_0}<\infty ,\mathbb{E}({|y|}^{\alpha +{\epsilon }_0}{a}^{\alpha +{\epsilon }_0})<\infty .\end{array}$$(2.12) ).

2 $loga$ is not supported by the set of the form $c_1+c_2\mathbb{Z}$, where $\mathbb{Z}$ is the set of integers and $c_1,c_2\in \mathbb{R}$ are fixed.

Additional information

Funding

The research was supported by the NCN [grant number UMO-2014/15/B/ST1/00060].