### Abstract

Original language | English |
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Place of Publication | Eindhoven |

Publisher | Technische Universiteit Eindhoven |

Number of pages | 12 |

Publication status | Published - 1993 |

### Publication series

Name | Memorandum COSOR |
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Volume | 9302 |

ISSN (Print) | 0926-4493 |

### Fingerprint

### Cite this

*Characterizations of shift-invariant distributions based on summation modulo one*. (Memorandum COSOR; Vol. 9302). Eindhoven: Technische Universiteit Eindhoven.

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*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.

Research output: Book/Report › Report › Academic

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 -