## Annals of Statistics

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

#### 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

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)

#### 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

#### References

• ABRAMOWITZ, M. and STEGUN, I. A. (1972). Handbook of Mathematical Functions. Dover, New York.
• DOy LE, P. G. and SNELL, J. L. (1984). Random Walks and Electric Networks. Math. Assoc. Amer., Washington, DC.
• DURRETT, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, Belmont, CA.
• EATON, M. L. (1992). A statistical dipty ch: Admissible inferences-recurrence of sy mmetric Markov chains. Ann. Statist. 20 1147-1179.
• EATON, M. L. (1997). Admissibility in quadratically regular problems and recurrence of sy mmetric Markov chains: Why the connection? J. Statist. Plann. Inference 64 231-247.
• HOBERT, J. P. and ROBERT, C. P. (1999). Eaton's Markov chain, its conjugate partner and -admissibility. Ann. Statist. 27 361-373.
• KNOPP, K. (1990). Theory and Application of Infinite Series. Dover, New York.
• LAMPERTI, J. (1960). Criteria for the recurrence or transience of stochastic processes, I. J. Math. Anal. Appl. 1 314-330.
• Ly ONS, T. (1983). A simple criterion for transience of a reversible Markov chain. Ann. Probab. 11 393-402.
• MCGUINNESS, S. (1991). Recurrent networks and a theorem of Nash-Williams. J. Theoret. Probab. 4 87-100.
• NASH-WILLIAMS, C. ST. J. A. (1959). Random walk and electric currents in networks. Proc. Cambridge Philosophical Soc. 55 181-194.
• PERES, Y. (1999). Probability on trees: An introductory climb. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1717 193-280. Springer, Berlin.
• GAINESVILLE, FLORIDA E-MAIL: jhobert@stat.ufl.edu DEPARTMENT OF MATHEMATICS CORNELL UNIVERSITY
• ITHACA, NEW YORK E-MAIL: jasonsch@poly gon.math. cornell.edu