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CHARACTERIZATIONS OF SHIFT‐INVARIANT DISTRIBUTIONS BASED ON SUMMATION MODULO ONE
Author(s) -
Wilms Roel J.G.,
Thiemann Jan G.F.
Publication year - 1994
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1994.tb00887.x
Subject(s) - modulo , mathematics , invariant (physics) , random variable , generality , lattice (music) , combinatorics , pure mathematics , discrete mathematics , physics , mathematical physics , statistics , psychology , acoustics , psychotherapist
Summary Let X 1 Y 1, …, Y n be independent random variables. We characterize the distributions of X and Y j satisfying the equation { X + Y 1 ++ Y n }= d X , where { Z } denotes the fractional part of a random variable Z. In the case of full generality, either X is uniformly distributed on [0,1), or Y j has.a shifted lattice distribution and X is shift‐invariant. We also give a characterization of shift‐invariant distributions. Finally, we consider some special cases of this equation.