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pith:ZHCXY6XI

pith:2026:ZHCXY6XIOTXSFVAEBQD2F5K4XM

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Revisiting Toeplitz and Hankel random matrices via $*$-convergence of circulant-type matrices

Arup Bose, Pradeep Vishwakarma

Random symmetric Toeplitz matrices converge in *-distribution to the sum of two non-commuting self-adjoint real Gaussian variables, while Hankel matrices converge to sums of symmetrized Rayleigh variables.

arxiv:2605.16160 v1 · 2026-05-15 · math.PR

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Claims

C1strongest claim

The random symmetric Toeplitz matrix converges in *-distribution to the sum of two non-commutative self-adjoint variables, each having a real Gaussian distribution. The random Hankel matrix converges to a sum of two self-adjoint variables each with symmetrized Rayleigh distribution.

C2weakest assumption

The derivation rests on the structural connections between circulant, reverse-circulant and left skew-circulant matrices and the Toeplitz/Hankel matrices, together with the joint *-convergence of a random circulant matrix and a deterministic diagonal matrix.

C3one line summary

The authors establish *-convergence of random Toeplitz and Hankel matrices to sums of Gaussian or Rayleigh non-commutative variables via circulant connections, yielding new proofs and moment expressions for their limiting spectral distributions.

References

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[1] Anderson, G.W., Guionnet, A. and Zeitouni, O. (2009).An Introduction to Random Matrices, Cambridge University Press, Cambridge 2009

[2] Adhikari, K. and Bose, A. (2019). Brown measure and asymptotic freeness of elliptic and related matrices.Random Matrix: Theory Appl.,8(2), 1950007 2019

[3] Bose, A., Hazra, R.S. and Saha, K. (2011). Convergence of joint moments for inde- pendent random patterned matrices.Ann. Probab.,39(4), 1607-1620 2011

[4] Basu, R., Bose, A., Ganguly, S. and Hazra, R.S. (2012). Joint convergence of several copies of different patterned random matrices.Electron. J. Probab.,17(82), 1-33 2012

[5] Bose, A. and Saha, K. (2018).Random Circulant Matrices, Chapman & Hall 2018

Receipt and verification

First computed	2026-05-20T00:01:55.515495Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c9c57c7ae874ef22d4040c07a2f55cbb072e1c988b1f430b162653e20c721fb8

Aliases

arxiv: 2605.16160 · arxiv_version: 2605.16160v1 · doi: 10.48550/arxiv.2605.16160 · pith_short_12: ZHCXY6XIOTXS · pith_short_16: ZHCXY6XIOTXSFVAE · pith_short_8: ZHCXY6XI

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ZHCXY6XIOTXSFVAEBQD2F5K4XM \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: c9c57c7ae874ef22d4040c07a2f55cbb072e1c988b1f430b162653e20c721fb8

Canonical record JSON

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  "metadata": {
    "abstract_canon_sha256": "feeb69260a2b355d77939af877b3e1656cbeb5af825fe4ef5c35d87d165f28fa",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-15T16:39:43Z",
    "title_canon_sha256": "3d0eb034e00835568dad97e4083b28390b7afc0d928df5f0997b63f6ba4ff650"
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  "source": {
    "id": "2605.16160",
    "kind": "arxiv",
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}