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April, 1975 A New Bound for Standard $p$-Functions
V. M. Joshi
Ann. Probab. 3(2): 346-352 (April, 1975). DOI: 10.1214/aop/1176996405


Let $p(t), t \in (0, \infty)$ be the standard $p$-function of a regenerative phenomenon as defined in Kingman's theory. Let $p(1) = M$ and $\min \{p(t), 0 \leqq t \leqq 1\} = m$. Griffeath (1973) has derived a new upper bound for $M$ for given $m$ by using the Kingman inequalities of order $\leqq 3$. Here Griffeath's result is generalized by using the Kingman equalities of order $\leqq n$. Further taking limits as $n \rightarrow \infty$ a new upper bound is obtained which is uniformly strictly superior to the present known upper bound. Thus a part of the uncharted region in the $M - m$ diagram becomes charted by being shown inaccessible. This gives also an improved upper bound for the constant $\nu_0$.


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V. M. Joshi. "A New Bound for Standard $p$-Functions." Ann. Probab. 3 (2) 346 - 352, April, 1975.


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

zbMATH: 0306.60041
MathSciNet: MR368157
Digital Object Identifier: 10.1214/aop/1176996405

Primary: 60J10
Secondary: 60J25 , 60K05

Keywords: $p$-function , Kingman inequalities , Markov transition function , regenerative phenomena

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 2 • April, 1975
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