I conjecture a sufficient condition for a distribution to be asymptotically exponential in a strong sense.
Roughly speaking, the idea is this. Suppose the "expected residual lifetime," $E[X-x|X≥x]$ is approximately constant for large $x$. Then, I believe that the conditional tail distribution is approximately exponential, in the sense of being stochastically dominated by an exponential and dominating a similar exponential. Formally:
Conjecture Given any random variable $X$ with support on $[0,∞)$. If $lim_{x→∞}E[X-x|X≥x]= \lambda ,$
then for all $ε>0$ and for all $\Delta>0$ there is some $c$ such that $x≥c$ implies $e^{-(1/(λ-ε))t}≥Pr[X≥x+t|X≥x]≥e^{-(1/(λ+ε))t} \qquad ∀t≥\Delta.$
Update 2 The first conjecture was wrong. Robert Israel provided a counterexample. The implication is now weaker, restricting $∀t≥\Delta>0$. The weakening takes care of the counterexample. But is it correct?
Update 3 (Removed)
Update 4 I posted the question on MathOverflow.
Update 5 The approximation result is stronger than weak convergence. Let $Y$ be distributed exponentially with parameter $\lambda$. The conclusion of the conjecture implies that
$\lim_{x→∞}\mathbb{E}[f(X-x)|X≥x]=\mathbb{E}[f(Y)]$
for all nondecreasing functions for which $\mathbb{E}[f(Y)]$ exists. In particular, $f$ is allowed to be unbounded.