Overview
This study examines matrix perpetuities, which are solutions to a specific type of affine fixed-point equation. The primary focus is on understanding the empirical spectral distribution of these solutions. The work addresses the existence and uniqueness of such solutions and explores their properties under conditions of orthogonal invariance. A key aspect involves establishing a relationship between matrix perpetuities and classical vector perpetuities.
Research Context
Matrix perpetuities are defined by the affine fixed-point equation:
$ \mathbf{X} \stackrel{d}{=} \mathbf{A}\,\mathbf{X} \,\mathbf{A}^\top+\mathbf{B},\qquad (\mathbf{A},\mathbf{B})\mbox{ and }\mathbf{X} \mbox{ are independent} $
The research investigates the behavior of the empirical spectral distribution of $\mathbf{X}$, the solution to this equation. The problem is initially framed by connecting it to classical vector perpetuities to establish foundational results concerning existence and uniqueness.
Approach
The research employs a multi-faceted approach. First, the problem of matrix perpetuities is related to classical vector perpetuities to demonstrate the existence and uniqueness of solutions. Under conditions of orthogonal invariance, a compression identity for symmetric multiplicative convolution is derived. This identity is utilized to show that principal submatrices of a matrix perpetuity also constitute lower-dimensional matrix perpetuities.
For models characterized by positive semidefinite and orthogonally invariant properties, the study develops a finite-dimensional spectral Kesten theorem. This theorem is used to obtain precise power-law tail asymptotics for the expected empirical spectral distribution. The tail behavior is shown to be governed by the largest eigenvalue, and its relationship to the tail of any diagonal entry is explicitly defined.
Furthermore, in the subcritical regime, the research demonstrates that the expected empirical spectral distribution of matrix perpetuities converges weakly to the distribution of the corresponding free perpetuity as the dimension tends to infinity.
Findings
- Existence and uniqueness results for matrix perpetuities were established by linking them to classical vector perpetuities.
- Under orthogonal invariance, a compression identity for symmetric multiplicative convolution was proven.
- It was shown that principal submatrices of a matrix perpetuity are themselves lower-dimensional matrix perpetuities.
- For positive semidefinite, orthogonally invariant models, a finite-dimensional spectral Kesten theorem was proven.
- This theorem yielded precise power-law tail asymptotics for the expected empirical spectral distribution.
- The tail of the expected empirical spectral distribution was found to be governed by the largest eigenvalue.
- The tail of the expected empirical spectral distribution was explicitly related to the tail of any diagonal entry.
- In the subcritical regime, the expected empirical spectral distribution of matrix perpetuities converges weakly to the distribution of the corresponding free perpetuity as the dimension approaches infinity.
Potential Applications
The findings are illustrated by matrix Beta prime perpetuities, for which explicit limiting spectral distributions are available.