I'm trying to understand the theory of "limits of statistical experiments" as explicated in Chapter 9 of Van Der Vaart's text, "Asymptotic Statistics". For some models, the limiting experiment is non-normal, such as when the experiments are uniform with non-common support. The key theorem is that they converge to an exponential experiment:
(The footnote is that $P_\theta$ may be defined arbitrarily for $\theta<0$.)
For certain statistics, such as the largest order statistic, this makes sense to me, it is known to have an exponential type of limit. But what if I am just looking at an average. For example in the $nth$ experiment, where the observations $X_1,\ldots,X_n$ are iid uniform on $(0,\theta-h/n)$, I take their average and normalize it, $\sqrt{n}(\overline{X}-\theta/2)$. (I believe this is a valid statistic in this setup because the parameter is technically $h$, not $\theta$, as he discusses in other examples.) This sequence of statistics should be asymptotically normal.
But then the limit should have a distribution in the limiting experiment:
Does it follow that the normal distribution can be represented as a randomized statistic in the exponential experiment, that is, as some function of a shifted exponentially distributed random variable $Z$ and an independent uniform random variable? I guess this is technically possible by ignoring $Z$ and applying $\Phi^{-1}$ to the uniform...But I still can't reconcile his remark quoted above that a "normal approximation is impossible". Or am I misunderstanding the theorems?
