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Convergence of patterned matrices with random walk entries

Arup Bose, Pradeep Vishwakarma

TL;DR

This work analyzes the convergence of high-dimensional patterned matrices whose entries are independent continuous-time random walks, establishing algebraic and joint convergence to the free Brownian motion within a non-commutative probability framework. It further introduces time-changed variants driven by inverse stable subordinators and demonstrates that CTRW-based matrices approximate these processes, though with heavier spectral tails under time-change. For select link functions, the authors derive explicit eigenvalue expressions that yield weak approximations to classical Brownian motion and to its time-changed counterpart, linking random matrix theory to anomalous diffusion models. Collectively, the results extend asymptotic freeness and connect patterned random matrices to time-changed free processes, enriching both the theory and potential applications in spectral analysis and stochastic modeling.

Abstract

It is well known that the Brownian motion on the real line can be obtained as a weak limit of a suitably scaled continuous-time random walk (CTRW). We investigate the convergence of certain patterned random matrices whose entries are independent CTRWs of various types. In a non-commutative probability framework, we use these high dimensional matrices to derive approximations of the free Brownian motion. Furthermore, we introduce and analyze a random time-changed version of the free Brownian motion driven by an inverse stable subordinator. An approximation of this process is obtained using a random matrix whose entries consist of continuous-time randomly stopped random walks. Moreover, it is shown that the empirical spectral distributions of such matrices have longer tails. Additionally, in a specific case, we use the explicit eigenvalue expressions of these matrices to obtain weak approximations of the standard Brownian motion and a time-changed variant of it.

Convergence of patterned matrices with random walk entries

TL;DR

This work analyzes the convergence of high-dimensional patterned matrices whose entries are independent continuous-time random walks, establishing algebraic and joint convergence to the free Brownian motion within a non-commutative probability framework. It further introduces time-changed variants driven by inverse stable subordinators and demonstrates that CTRW-based matrices approximate these processes, though with heavier spectral tails under time-change. For select link functions, the authors derive explicit eigenvalue expressions that yield weak approximations to classical Brownian motion and to its time-changed counterpart, linking random matrix theory to anomalous diffusion models. Collectively, the results extend asymptotic freeness and connect patterned random matrices to time-changed free processes, enriching both the theory and potential applications in spectral analysis and stochastic modeling.

Abstract

It is well known that the Brownian motion on the real line can be obtained as a weak limit of a suitably scaled continuous-time random walk (CTRW). We investigate the convergence of certain patterned random matrices whose entries are independent CTRWs of various types. In a non-commutative probability framework, we use these high dimensional matrices to derive approximations of the free Brownian motion. Furthermore, we introduce and analyze a random time-changed version of the free Brownian motion driven by an inverse stable subordinator. An approximation of this process is obtained using a random matrix whose entries consist of continuous-time randomly stopped random walks. Moreover, it is shown that the empirical spectral distributions of such matrices have longer tails. Additionally, in a specific case, we use the explicit eigenvalue expressions of these matrices to obtain weak approximations of the standard Brownian motion and a time-changed variant of it.

Paper Structure

This paper contains 17 sections, 4 theorems, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Non-commutative probability and (free) independence
  4. Independence and free independence
  5. Joint distribution and asymptotic freeness
  6. Free Brownian motion
  7. Patterned random matrices and link function
  8. Property B
  9. Joint convergence of independent random matrices
  10. Inverse stable subordinator and fractional Poisson process
  11. Patterned random matrices with CTRW entries
  12. Marginal convergence
  13. Joint convergence
  14. Symmetric circulant matrix and SBM
  15. Time-changed free Brownian motion
  16. ...and 2 more sections

Key Result

Corollary 3.1

Let $\{X_{i,j,k},\ 1\leq i\leq j\leq n,\ k\ge1\}$, $n\in\mathbb{N}$ be a triple indexed sequence that satisfies Assumption I. Let $\{\{A_n(t)\}_{t\ge0}\}_{n\ge1}$ be a sequence of matrix valued random processes as defined in (matrixdef) with link function $L$. Then, (i) for $L$ defined by $L(i,j)=(\

Figures (2)

  • Figure 1: ESD of $n^{-1/2}A_n^\alpha(t)$ for Wigner type link function, $n=1000$, $t=1$, and with standard Gaussian step variables $X_{i,j,k}$'s.
  • Figure 2: ESD of $n^{-1/2}A_n^\alpha(t)$ for Symmetric Circulant type link function, $n=1500$, $t=1$, and with standard Gaussian step variables $X_{i,j,k}$'s.

Theorems & Definitions (11)

  • Corollary 3.1
  • proof
  • proof
  • Proposition 4.1
  • proof
  • proof
  • Lemma 4.1
  • proof
  • proof
  • Corollary 4.1
  • ...and 1 more