Tails of bivariate stochastic recurrence equation with triangular matrices

TL;DR

This paper analyzes the tail behavior of stationary solutions to bivariate stochastic recurrence equations with triangular matrices, revealing novel asymptotics where components have different tail indices, extending previous results.

Contribution

It provides a complete characterization of tail behavior for bivariate stochastic recurrence equations with triangular matrices under standard conditions, uncovering new tail asymptotics.

Findings

W_1 and W_2 can have different regularly varying tail indices

Complete tail asymptotic characterization under Kesten-Goldie and Grey conditions

Novel tail behavior not observed in previous stochastic recurrence models

Abstract

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.