Characterizations of shift-invariant distributions based on summation modulo one

R.J.G. Wilms, J.G.F. Thiemann

Research output: Book/ReportReportAcademic

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Abstract

For n \in N, let X, Y_1, ..., Y_n be independent random variables, and suppose that X is distributed in [0,1), but not uniformly. We characterize the distributions of X and Y_s (s=1,...,n) satisfying the equation $\{ X+Y_1+...+Y_n\} \stackrel{\rm{d}}{=} X$, where {Z} denotes the fractional part of a random variable Z. In the case of full generality, Y_s is lattice, and X is shift-invariant with respect to a discrete unifonn distribution on [0,1). We also give a characterization of such shift-invariant distributions. In addition, we consider some special cases of this equation: If $X \stackrel{\rm{d}}{=} Y_1$, then X has a shifted discrete uniform distribution on [0,1); further the case that Y_1, ..., Y_n are identically distributed, and a generalization of the equation with X, Y_1, ..., Y_n identically distributed is considered. Our results generalize results of Goldman (1968) and of Arnold and Meeden (1976). Key words and phrases: Fourier-Stieltjes coefficients; distribution modulo 1; fractional parts.
Original languageEnglish
Place of PublicationEindhoven
PublisherTechnische Universiteit Eindhoven
Number of pages12
Publication statusPublished - 1993

Publication series

NameMemorandum COSOR
Volume9302
ISSN (Print)0926-4493

Fingerprint

Invariant Distribution
Summation
Fractional Parts
Modulo
Identically distributed
Distribution Modulo 1
Discrete uniform distribution
Discrete Distributions
Independent Random Variables
Random variable
Denote
Generalise
Invariant
Coefficient

Cite this

Wilms, R. J. G., & Thiemann, J. G. F. (1993). Characterizations of shift-invariant distributions based on summation modulo one. (Memorandum COSOR; Vol. 9302). Eindhoven: Technische Universiteit Eindhoven.
Wilms, R.J.G. ; Thiemann, J.G.F. / Characterizations of shift-invariant distributions based on summation modulo one. Eindhoven : Technische Universiteit Eindhoven, 1993. 12 p. (Memorandum COSOR).
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abstract = "For n \in N, let X, Y_1, ..., Y_n be independent random variables, and suppose that X is distributed in [0,1), but not uniformly. We characterize the distributions of X and Y_s (s=1,...,n) satisfying the equation $\{ X+Y_1+...+Y_n\} \stackrel{\rm{d}}{=} X$, where {Z} denotes the fractional part of a random variable Z. In the case of full generality, Y_s is lattice, and X is shift-invariant with respect to a discrete unifonn distribution on [0,1). We also give a characterization of such shift-invariant distributions. In addition, we consider some special cases of this equation: If $X \stackrel{\rm{d}}{=} Y_1$, then X has a shifted discrete uniform distribution on [0,1); further the case that Y_1, ..., Y_n are identically distributed, and a generalization of the equation with X, Y_1, ..., Y_n identically distributed is considered. Our results generalize results of Goldman (1968) and of Arnold and Meeden (1976). Key words and phrases: Fourier-Stieltjes coefficients; distribution modulo 1; fractional parts.",
author = "R.J.G. Wilms and J.G.F. Thiemann",
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language = "English",
series = "Memorandum COSOR",
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Wilms, RJG & Thiemann, JGF 1993, Characterizations of shift-invariant distributions based on summation modulo one. Memorandum COSOR, vol. 9302, Technische Universiteit Eindhoven, Eindhoven.

Characterizations of shift-invariant distributions based on summation modulo one. / Wilms, R.J.G.; Thiemann, J.G.F.

Eindhoven : Technische Universiteit Eindhoven, 1993. 12 p. (Memorandum COSOR; Vol. 9302).

Research output: Book/ReportReportAcademic

TY - BOOK

T1 - Characterizations of shift-invariant distributions based on summation modulo one

AU - Wilms, R.J.G.

AU - Thiemann, J.G.F.

PY - 1993

Y1 - 1993

N2 - For n \in N, let X, Y_1, ..., Y_n be independent random variables, and suppose that X is distributed in [0,1), but not uniformly. We characterize the distributions of X and Y_s (s=1,...,n) satisfying the equation $\{ X+Y_1+...+Y_n\} \stackrel{\rm{d}}{=} X$, where {Z} denotes the fractional part of a random variable Z. In the case of full generality, Y_s is lattice, and X is shift-invariant with respect to a discrete unifonn distribution on [0,1). We also give a characterization of such shift-invariant distributions. In addition, we consider some special cases of this equation: If $X \stackrel{\rm{d}}{=} Y_1$, then X has a shifted discrete uniform distribution on [0,1); further the case that Y_1, ..., Y_n are identically distributed, and a generalization of the equation with X, Y_1, ..., Y_n identically distributed is considered. Our results generalize results of Goldman (1968) and of Arnold and Meeden (1976). Key words and phrases: Fourier-Stieltjes coefficients; distribution modulo 1; fractional parts.

AB - For n \in N, let X, Y_1, ..., Y_n be independent random variables, and suppose that X is distributed in [0,1), but not uniformly. We characterize the distributions of X and Y_s (s=1,...,n) satisfying the equation $\{ X+Y_1+...+Y_n\} \stackrel{\rm{d}}{=} X$, where {Z} denotes the fractional part of a random variable Z. In the case of full generality, Y_s is lattice, and X is shift-invariant with respect to a discrete unifonn distribution on [0,1). We also give a characterization of such shift-invariant distributions. In addition, we consider some special cases of this equation: If $X \stackrel{\rm{d}}{=} Y_1$, then X has a shifted discrete uniform distribution on [0,1); further the case that Y_1, ..., Y_n are identically distributed, and a generalization of the equation with X, Y_1, ..., Y_n identically distributed is considered. Our results generalize results of Goldman (1968) and of Arnold and Meeden (1976). Key words and phrases: Fourier-Stieltjes coefficients; distribution modulo 1; fractional parts.

M3 - Report

T3 - Memorandum COSOR

BT - Characterizations of shift-invariant distributions based on summation modulo one

PB - Technische Universiteit Eindhoven

CY - Eindhoven

ER -

Wilms RJG, Thiemann JGF. Characterizations of shift-invariant distributions based on summation modulo one. Eindhoven: Technische Universiteit Eindhoven, 1993. 12 p. (Memorandum COSOR).