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June, 1975 Generalized Distribution Functions: The $\sigma$-Lower Finite Case
Gordon Simons
Ann. Probab. 3(3): 492-502 (June, 1975). DOI: 10.1214/aop/1176996355


A mass $m(x) \geqq 0$ is assigned to each point $x$ of a partially ordered countable set $X$. It is further assumed that $M(x) = \sum_{y\leqq x} m(y) < \infty$ for each $x \in X. M$ is called a distribution function. For certain sets $X$, it is shown that $M$ determines $m$. For others, $M$ need not determine $m$ uniquely. A theory is presented for $\sigma$-lower finite spaces (sets), which are defined in the paper. Such spaces are locally finite. That is, each interval $\lbrack x, y \rbrack = \{z \in X: x \leqq z \leqq y\}$ has a finite number of points. Mobius functions, which have been defined for locally finite spaces, are used throughout. Distribution functions on a particular $\sigma$-lower finite space arise naturally from boundary crossing problems analyzed by Doob and Anderson. The theory is applied to this example and to another.


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Gordon Simons. "Generalized Distribution Functions: The $\sigma$-Lower Finite Case." Ann. Probab. 3 (3) 492 - 502, June, 1975.


Published: June, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0312.60007
MathSciNet: MR375431
Digital Object Identifier: 10.1214/aop/1176996355

Primary: 60E05
Secondary: 06A10 , 60G17

Keywords: Boundary crossing , distribution function , Mobius function , partial ordering

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 3 • June, 1975
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