Hidden regular variation for stochastic recursions with diagonal matrices

TL;DR

This work analyzes multivariate tails for stationary solutions ${\boldsymbol X}$ of the stochastic recurrence ${\boldsymbol X}= {\boldsymbol A}{\boldsymbol X}+{\boldsymbol B}$ with a random diagonal matrix ${\boldsymbol A}$. It builds a framework of assumptions and notation to establish hidden regular variation on ${\mathbb R}^2 \setminus [\mathsf{axes}]$, identifying a second-order scaling $(\log t)^{1/2} t^{\xi_1^*+\xi_2^*}$ and a nontrivial limit measure ${\Lambda}_2$ when an appropriate Crámer-type condition holds and ${\boldsymbol\xi}^*$ maximizes the growth subject to ${\varphi({\boldsymbol\xi})=1}$ with $\nabla\varphi({\boldsymbol\xi}^*)$ parallel to $(1,1)$. The paper develops a two-dimensional implicit renewal theory, proves moment bounds, and analyzes a two-block structure to extend the phenomenon to higher dimensions; it demonstrates hidden regular variation in concrete diagonal models such as CCC-GARCH, diagonal BEKK-ARCH, and a log-Gaussian SRE, providing precise joint-tail probabilities and tail measures. These results enhance understanding of joint extremes in multivariate stochastic recursions and have potential applications in risk management where simultaneous large moves matter.

Abstract

We consider random vectors $X$ that satisfy the equation in law $X=AX+B$, where $A$ is a given random diagonal matrix and $B$ a given random vector, both independent of $X$. It is well known by the works of Kesten and Goldie that the marginals of $X$ may exhibit heavy tails, with possibly different tail indices. In recent works (Damek 2025, Mentemeier and Wintenberger 2022) it was observed that asymptotic independence may occur despite strong dependencies in the entries of $A$: The probability that both marginals are simultaneously large decays faster than the marginal probability of an extreme event; the tail measure is concentrated on the axis. In this work, we analyse the hidden regular variation properties of $X$, that is, we find the proper scaling for which one observes simultaneous extremes.

Figures (3)

• Figure 1: Plots of the level set $D$. Left: \ref{['assump:positivedependence']} fails, Right: \ref{['assump:positivedependence']} holds.
• Figure 2: Plots of $\varphi$ for different parameters in the CCC-GARCH-model
• Figure 3: Plots of $\boldsymbol \Phi(\boldsymbol \theta):=\mathop{\mathrm{\mathds{E}}}\nolimits[ |A_1|^{\theta_1} |A_2|^{\theta_2} ]$ for different parameters in the BEKK-ARCH(2)-model

Theorems & Definitions (52)

• Lemma 3.1
• Remark 3.2
• Proposition 3.3
• Theorem 3.4
• Remark 3.5
• Theorem 3.6
• Remark 3.7
• proof : Proof of Lemma \ref{['lem:properties.phi2']}
• proof : Proof of Proposition \ref{['prop:m-delta']}
• Theorem 6.1