{
  "id": "2005.02915",
  "version": "v1",
  "published": "2020-05-06T15:45:41.000Z",
  "updated": "2020-05-06T15:45:41.000Z",
  "title": "An almost sure invariance principle for some classes of inhomogeneous Markov chains and non-stationary $ρ$-mixing sequences",
  "authors": [
    "Yeor Hafouta"
  ],
  "comment": "14 pp",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove a vector-valued almost sure invariance principle for partial sums generated by uniformly contracting or elliptic Markov chains and a uniformly bounded sequence of functions. In the real-valued case we will also consider other types of non-stationary $\\rho$-mixing sequences. In the scalar case, when the variance $\\sig_n^2$ of the underlying partial sums $S_n$ grows at least as fast as $n^\\ve$ (for some $\\ve>0$), we obtain the rate $\\sig_n^{1/2+\\del}$ for any $\\del>0$, while in the vector-valued case, for sufficiently regular functions, we obtain the rate $s_n^{1/2+\\del}$, where $s_n^2=\\min_{|v|=1}v\\cdot \\text{Cov}(S_n)v$ is the \"growth rate\" of the of covariance matrix of $S_n$ in the space of positive definite matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-06T15:45:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sure invariance principle",
      "inhomogeneous markov chains",
      "mixing sequences",
      "non-stationary",
      "partial sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}