{
  "id": "2209.12535",
  "version": "v1",
  "published": "2022-09-26T09:31:56.000Z",
  "updated": "2022-09-26T09:31:56.000Z",
  "title": "Almost sure invariance principle of $β-$mixing time series in Hilbert space",
  "authors": [
    "Jianya Lu",
    "Wei Biao Wu",
    "Zhijie Xiao",
    "Lihu Xu"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Inspired by \\citet{Berkes14} and \\citet{Wu07}, we prove an almost sure invariance principle for stationary $\\beta-$mixing stochastic processes defined on Hilbert space. Our result can be applied to Markov chain satisfying Meyn-Tweedie type Lyapunov condition and thus generalises the contraction condition in \\citet[Example 2.2]{Berkes14}. We prove our main theorem by the big and small blocks technique and an embedding result in \\citet{gotze2011estimates}. Our result is further applied to the ergodic Markov chain and functional autoregressive processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-26T09:31:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sure invariance principle",
      "mixing time series",
      "hilbert space",
      "meyn-tweedie type lyapunov condition",
      "chain satisfying meyn-tweedie type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}