Remarks on characterization of normal and stable distributions

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Let X, Y, Z be independent identically distributed (i.i.d.) random variables. Suppose {Mathematical expression} for all real t, u, v, where q=2 and p≠2m (m=1, 2,...) or 0<p<q<2. It was proved by the author this implies X, Y, Z have the symmetric q-stable distribution. For two random variables such result is not true. One may suppose that the condition {Mathematical expression} and additional assumption on the behavior of P{|X|≥x} (x→∞) imply X, Y are stable. In this paper we show it is not valid. The second result is: if the last relation holds for two different exponents and q=2, then X and Y are normal.

Original languageEnglish
Pages (from-to)407-415
Number of pages9
JournalJournal of Theoretical Probability
Issue number2
StatePublished - 1 Apr 1993
Externally publishedYes


  • Distribution
  • characterization
  • function
  • random variables

ASJC Scopus subject areas

  • Statistics and Probability
  • General Mathematics
  • Statistics, Probability and Uncertainty


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