Limit theorems for transient diffusions on the line (Q756275)

Let X be a diffusion in natural scale on (0,1], with 1 reflecting, and let \(c(x)\equiv {\mathbb{E}}(H_ x)\) and \(v(x)\equiv var(H_ x)\), where \(H_ x=\inf \{t:\;X_ t=x\}.\) Let \(\sigma_ x=\sup \{t:\;X_ t=x\}.\) The main results of this paper are firstly that (i) c is slowly varying; (ii) \(c(X_ t)/t\to^{{\mathbb{P}}}1;\) (iii) \(H_ x/c(x)\to^{{\mathbb{P}}}1;\) (iv) \(\sigma_ x/c(x)\to^{{\mathbb{P}}}1\) are all equivalent; and secondly that (v) \(c(X_ t)/t\to^{a.s.}1;\) (vi) \(H_ x/c(x)\to^{a.s.}1;\) (vii) \(\sigma_ x/c(x)\to^{a.s.}1\) are all equivalent, and are implied by the condition \(\int_{0+}c(x)^{-2}dv(x)<\infty.\) Other partial results for more general limit theorems are proved, and new results on regular variation are established.