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 $\nu_0$.
@article{1176996405,
author = {Joshi, V. M.},
title = {A New Bound for Standard $p$-Functions},
journal = {Ann. Probab.},
volume = {3},
number = {6},
year = {1975},
pages = { 346-352},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996405}
}
Joshi, V. M. A New Bound for Standard $p$-Functions. Ann. Probab., Tome 3 (1975) no. 6, pp. 346-352. http://gdmltest.u-ga.fr/item/1176996405/