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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Abstract

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

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

Dates
First available in Project Euclid: 10 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1031689024

Digital Object Identifier
doi:10.1214/aos/1031689024

Mathematical Reviews number (MathSciNet)
MR1926175

Zentralblatt MATH identifier
1103.60315

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

Keywords
Admissibility electrical network geometric distribution null recurrence reversibility weighted random walk

Citation

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. https://projecteuclid.org/euclid.aos/1031689024


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  • GAINESVILLE, FLORIDA E-MAIL: jhobert@stat.ufl.edu DEPARTMENT OF MATHEMATICS CORNELL UNIVERSITY
  • ITHACA, NEW YORK E-MAIL: jasonsch@poly gon.math. cornell.edu