The Annals of Statistics

Conditions for recurrence and transience of a Markov chain on $\mathbb{Z}^+$ and estimation of a geometric success probability

James P. Hobert and Jason Schweinsberg

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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.

Article information

Ann. Statist., Volume 30, Number 4 (2002), 1214-1223.

First available in Project Euclid: 10 September 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 62C15: Admissibility

Admissibility electrical network geometric distribution null recurrence reversibility weighted random walk


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. 30 (2002), no. 4, 1214--1223. doi:10.1214/aos/1031689024.

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