Let $Z$ be a discrete random variable with support $\Z^+ = \{0,1,2,\dots\}$. We consider a Markov chain $Y=(Y_n)_{n=0}^\infty$ with state space $\Z^+$ and transition probabilities given by $P(Y_{n+1} = j|Y_n = i) = P(Z = i+j)/P(Z \geq i)$. We prove that convergence of $\sum_{n=1}^\infty 1/[n^3 P (Z=n)]$ is sufficient for transience of $Y$ while divergence of $\sum_{n=1}^\infty 1/[n^2 P (Z \geq n)]$ is sufficient for recurrence. Let $X$ be a $\mbox{Geometric}(p)$ random variable; that is, $P(X=x)=p(1-p)^x$ for $x \in \Z^+$. We use our results in conjunction with those of M. L. Eaton [Ann. Statist.
20 (1992) 1147-1179] and J. P. Hobert and C. P. Robert [Ann. Statist.
27 (1999) 361-373] to establish a sufficient condition for $\mathscr{P}$-admissibility of improper priors on $p$. As an illustration of this result, we prove that all prior densities of the form $p^{-1}(1-p)^{b-1}$ with $b>0$ are $\mathscr{P}$-admissible.
Publié le : 2002-08-14
Classification:
Admissibility,
electrical network,
geometric distribution,
null recurrence,
reversibility,
weighted random walk,
60J10,
62C15
@article{1031689024,
author = {Hobert, James P. and Schweinsberg, Jason},
title = {Conditions for recurrence and transience of a Markov chain on $\mathbb{Z}^+$ and estimation of a geometric success probability},
journal = {Ann. Statist.},
volume = {30},
number = {1},
year = {2002},
pages = { 1214-1223},
language = {en},
url = {http://dml.mathdoc.fr/item/1031689024}
}
Hobert, James P.; Schweinsberg, Jason. Conditions for recurrence and transience of a Markov chain on $\mathbb{Z}^+$ and estimation of a geometric success probability. Ann. Statist., Tome 30 (2002) no. 1, pp. 1214-1223. http://gdmltest.u-ga.fr/item/1031689024/