Density estimation from an individual numerical sequence

Andrew B. Nobel, Gusztáv Morvai, Sanjeev R. Kulkarni

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


This paper considers estimation of a univariate density from an individual numerical sequence. It is assumed that 1) the limiting relative frequencies of the numerical sequence are governed by an unknown density, and 2) there is a known upper bound for the variation of the density on an increasing sequence of intervals. A simple estimation scheme is proposed, and is shown to be L 1 consistent when 1) and 2) apply. In addition, it is shown that there is no consistent estimation scheme for the set of individual sequences satisfying only condition 1).

Original languageEnglish (US)
Pages (from-to)537-541
Number of pages5
JournalIEEE Transactions on Information Theory
Issue number2
StatePublished - 1998

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


  • Bounded variation
  • Density estimation
  • Ergodic processes
  • Individual sequences


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