Stability of the tail Markov chain and the evaluation of improper priors for an exponential rate parameter

Abstract

Let Z be a continuous random variable with a lower semicontinuous density f that is positive on (0,∞) and 0 elsewhere. Put G(x) = ∨_x^∞ f(z)dz. We study the tail Markov chain generated by Z, defined as the Markov chain Ψ=(Ψ_n)_n=0^∞ with state space [0, ∞) and Markov transition density k(y|x) = f(y+x)/G(x). This chain is irreducible, aperiodic and reversible with respect to G. It follows that Ψ is positive recurrent if and only if Z has a finite expectation. We prove (under regularity conditions) that if E Z = ∞, then Ψ is null recurrent if and only if ∨₁^∞ 1/[ z³ f(z) ] dz = ∞. Furthermore, we describe an interesting decision-theoretic application of this result. Specifically, suppose that X is an Exp(θ) random variable; that is, X has density θe^{- θx} for x>0. Let ν be an improper prior density for θ that is positive on (0,∞). Assume that ∨₀^∞ θ ν(θ) dθ< ∞, which implies that the posterior density induced by ν is proper. Let m_ν denote the marginal density of X induced by ν; that is, m_ν(x) = ∨₀^∞ θe^-θx ν(θ) dθ. We use our results, together with those of Eaton and of Hobert and Robert, to prove that ν is a \cal P-admissible prior if ∨₁^∞ 1/ [x² m_ν(x)]dx = ∞.