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, for some $\lambda \in(0,∞)$, $$lim_{x→∞}E[X-x|X≥x]= \lambda ,$$

then, for all $ε>0$ and for all $\Delta>0$, there is some $c$ such that $x≥c$ implies  $$e^{-(1/(λ-ε))t}\leq Pr[X≥x+t|X≥x] \leq e^{-(1/(λ+ε))t}    \qquad ∀t≥\Delta.$$


I posted this question on [StackExchange](http://math.stackexchange.com/questions/136489/sufficient-condition-for-asymptotically-exponential-tail-corrected-but-still-un). Robert Israel provided a counterexample to an earlier conjecture, which was wrong.

**Update** The approximation result is stronger than weak convergence. Let $Y$ be distributed exponentially with parameter $\lambda$. The conclusion of the conjecture implies that

$$lim_{x→∞}E[f(X-x)|X≥x]=E[f(Y)]$$

for all nondecreasing functions for which $E[f(Y)]$ exists. In particular, $f$ is allowed to be *unbounded*.