{
  "id": "2404.09525",
  "version": "v1",
  "published": "2024-04-15T07:42:14.000Z",
  "updated": "2024-04-15T07:42:14.000Z",
  "title": "Coupling results and Markovian structures for number representations of continuous random variables",
  "authors": [
    "Jesper Møller"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "A general setting for nested subdivisions of a bounded real set into intervals defining the digits $X_1,X_2,...$ of a random variable $X$ with a probability density function $f$ is considered. Under the weak condition that $f$ is almost everywhere lower semi-continuous, a coupling between $X$ and a non-negative integer-valued random variable $N$ is established so that $X_1,...,X_N$ have an interpretation as the ``sufficient digits'', since the distribution of $R=(X_{N+1},X_{N+2},...)$ conditioned on $S=(X_1,...,X_N)$ does not depend on $f$. Adding a condition about a Markovian structure of the lengths of the intervals in the nested subdivisions, $R\\,|\\,S$ becomes a Markov chain of a certain order $s\\ge0$. If $s=0$ then $X_{N+1},X_{N+2},...$ are IID with a known distribution. When $s>0$ and the Markov chain is uniformly geometric ergodic, a coupling is established between $(X,N)$ and a random time $M$ so that the chain after time $\\max\\{N,s\\}+M-s$ is stationary and $M$ follows a simple known distribution. The results are related to several examples of number representations generated by a dynamical system, including base-$q$ expansions, generalized L\\\"uroth series, $\\beta$-expansions, and continued fraction representations. The importance of the results and some suggestions and open problems for future research are discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-15T07:42:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuous random variables",
      "number representations",
      "markovian structure",
      "coupling results",
      "markov chain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}