{
  "id": "1707.08827",
  "version": "v1",
  "published": "2017-07-27T12:15:02.000Z",
  "updated": "2017-07-27T12:15:02.000Z",
  "title": "On the average of probability distributions of a discrete Markov chain",
  "authors": [
    "Nikolaos Halidias"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X_n$ be a discrete time Markov chain with state space $S$ and initial probability distribution $\\mu^{(0)} = (P(X_0=i_1),P(X_0=i_2),\\cdots,)$. What is the probability of choosing in random some $k \\in \\mathbb{N}$ with $k \\leq n$ such that $X_k = j$ where $j \\in S$? This probability is the average $\\frac{1}{n} \\sum_{k=1}^n \\mu^{(k)}_j$ where $\\mu^{(k)}_j = P(X_k = j)$. In this note we will study the limit of this average without assuming that the chain is irreducible. Finally, we study the limit of the average $\\frac{1}{n} \\sum_{k=1}^n g(X_k)$ where $g$ is a given function for a general Markov chain not necessarily irreducible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-27T12:15:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10"
    ],
    "keywords": [
      "discrete markov chain",
      "discrete time markov chain",
      "initial probability distribution",
      "general markov chain",
      "state space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}