TITLE: If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense
QUESTION [6 upvotes]: Suppose the "mean residual lifetime," $\mathbb{E}[X-x|X≥x]$ 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 $\lambda \in(0,∞)$, $$\lim_{x→∞}\mathbb{E}[X-x|X≥x]= \lambda ,$$
then, for all $ε>0$ and for all $\Delta>0$, there is some $c$ such that $x≥c$ implies  $$e^{-\frac{t}{λ-ε}}\leq \mathbb{P}[X≥x+t|X≥x] \leq e^{-\frac{t}{λ+ε}}    \qquad ∀t≥\Delta.$$
I posted this question on StackExchange. 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
$$\lim_{x→∞}\mathbb{E}[f(X-x)|X≥x]=\mathbb{E}[f(Y)]$$
for all nondecreasing functions for which $\mathbb{E}[f(Y)]$ exists. In particular, $f$ is allowed to be unbounded.  The first great response by Ori Gurel-Gurevich implies a slightly weaker approximation result.

REPLY [2 votes]: Here is a work which gives general conditions under which your conjecture is true.
"Limiting Properties of the Mean Residual Lifetime Function" by Isaac Meilijson in Ann. Math. Statist. Volume 43, Number 1 (1972), 354-357.
http://projecteuclid.org/euclid.aoms/1177692731