{
  "id": "1111.1901",
  "version": "v1",
  "published": "2011-11-08T13:22:53.000Z",
  "updated": "2011-11-08T13:22:53.000Z",
  "title": "Limiting Spectral Distribution of Block Matrices with Toeplitz Block Structure",
  "authors": [
    "Riddhipratim Basu",
    "Arup Bose",
    "Shirshendu Ganguly",
    "Rajat Subhra Hazra"
  ],
  "comment": "12 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study two specific symmetric random block Toeplitz (of dimension $k \\times k$) matrices: where the blocks (of size $n \\times n$) are (i) matrices with i.i.d. entries, and (ii) asymmetric Toeplitz matrices. Under suitable assumptions on the entries, their limiting spectral distributions (LSDs) exist (after scaling by $\\sqrt{nk}$) when (a) $k$ is fixed and $n \\to\\infty$ (b) $n$ is fixed and $k\\rightarrow \\infty$ (c) $n$ and $k$ go to $\\infty$ simultaneously. Further the LSD's obtained in (a) and (b) coincide with those in (c) when $n$ or respectively $k$ tends to infinity. This limit in (c) is the semicircle law in case (i). In Case (ii) the limit is related to the limit of the random symmetric Toepiltz matrix as obtained by Bryc et al.(2006) and Hammond and Miller(2005).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-08T13:22:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "limiting spectral distribution",
      "toeplitz block structure",
      "block matrices",
      "specific symmetric random block toeplitz",
      "random symmetric toepiltz matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.1901B"
    }
  }
}