Suppose random variable $X$ has a continuous probability distribution with an unbounded upper tail; that is, the CDF of $X$ (call it $F$) is absolutely continuous and $F(x)<1$ for all $x\in\mathbb{R}.$
Now consider the conditional CDF of ${X-a\over w}$ given that $a\le X\le a+w$, for $w>0$:$$G(y,a,w):=\mathbb{P}\left({X-a\over w}\le y\ {\LARGE \mid}\ a\le X\le a+w\right) ={F(a+w\,y)-F(a)\over F(a+w)-F(a)}\,1_{0\le y\le 1}+1_{y>1}.$$
WolframCloud computations lead to the following ...
Observation: Apparently, for all well-known parametric families of such distributions (e.g., Normal, Lognormal, Student's t, Cauchy, Maxwell, Gamma, Gumbel, Weibull, etc.), there are only three types of limit of the CDF of ${X-a\over w}\mid a\le X\le a+w$ as $a\to\infty$: $$\lim_{a\to\infty}G(y,a,w)=\lim_{a\to\infty}\mathbb{P}\left({X-a\over w}\le y\ {\LARGE \mid}\ a\le X\le a+w\right) \\[3ex] = \begin{cases} 1_{y>0} & \text{(Type 1: Degenerate at $0$)}\\[2ex] y\,1_{0\le y\le 1}+1_{y>1} & \text{(Type 2: Uniform on $[0,1]$)}\\[2ex] {1- e^{-y\,w/\beta}\over 1-e^{-w/\beta}}1_{0\le y\le 1}+1_{y>1} & \text{(Type 3: Exponential$(\beta/w)$ on $[0,1]$)} \end{cases}$$ where $\beta$ is a parameter depending on the $X$ distribution.
NB: The Type 3 limit depends on both $w$ and the $X$ distribution, but the Type 1 and Type 2 limits are free of w and free of all parameters of the $X$ distribution, which seems quite remarkable.
NB: Letting $Y_a={X-a\over w}$, for all three types (and any other type, if there are any) we have $$\begin{align}y\le 0&\implies \{Y_a\le y\}\cap\{0\le Y_a\le 1\}=\{Y_a=0\}\implies \lim_{a\to\infty}G(y,a,w)=0\\ y\ge 1&\implies \{Y_a\le y\}\cap\{0\le Y_a\le 1\}=\{0\le Y_a\le 1\}\implies \lim_{a\to\infty}G(y,a,w)=1\end{align}$$ since $\mathbb{P}(Y_a=0)=0$ due to the distribution of $X$ (hence $Y_a$) being continuous; thus, the types of limit differ only for $y$ in the open interval $(0,1)$.
Here are pictures of these three types of limit:
Questions:
How generally does this observation hold, and how can it be proved analytically? (Is this something well-known?)
What $X$ distributions (if any) do in fact yield a limit not among these three types?
In the Degenerate case, the limit is not right-continuous at $y=0$, so it fails to be a CDF. Nevertheless, as $a\to\infty$ the probability mass clearly concentrates in an arbitrarily small neighborhood of $y=0$ (as indicated by the plots below); so, is there a valid interpretation of this as a limiting distribution?
(My original motivating question concerned only the asymptotic variance for just Normal vs. Lognormal. These variances of course follow from the above limiting distributions.)
Examples
Degenerate Limit
Here are the CDFs (left) and corresponding PDFs (right) for Normal($0,1$), showing the CDFs, PDFs labeled as $G(y,a,w),\ g(y,a,w)$ for various $a$ with $w=1$: Computations show that the limit of the distribution is degenerate at $0$ for apparently any Normal, Gumbel, Rayleigh or Maxwell distribution, or any Weibull distribution with shape parameter $> 1.$
Uniform Limit
Here are the CDFs (left) and corresponding PDFs (right) for LogNormal(0,1), showing the CDFs, PDFs labeled as $G(y,a,w),\ g(y,a,w)$ for various $a$ with $w=1$: Computations show that the limit distribution is Uniform on $[0,1]$ for apparently any LogNormal, Student-t, Cauchy, Levi, Pareto, or Inverse Gamma distribution, or any Weibull distribution with shape parameter $< 1.$
Exponential Limit
Here are the CDFs (left) and corresponding PDFs (right) for Gamma($\alpha=2,\beta=3$), showing the CDFs, PDFs labeled as $G(y,a,w),\ g(y,a,w)$ for various $a$ with $w=1$:
Computations show that the limit distribution is Exponential on $[0,1]$ for apparently any Gamma distribution (which includes the Weibull distribution with shape parameter $= 1$), or any Logistic, Laplace, or Extreme Value distribution.
For reference, here's the Wolfram code I used:
(* Find limit(a->infinity) Pr[(X-a)/w <= y | a<=X<=a+w] for 0<y<1 *)
FindLimit[distr_,asms_]:=Module[{assms=asms~Join~{0<y<1,a>0,w>0}},
G[y_,a_,w_]:=Probability[(X-a)/w<=y\[Conditioned]a<=X<=a+w,X\[Distributed]distr];
Glim[y_,w_]:=Simplify[Cancel[Limit[G[y,a,w],a->Infinity]]];
Print["0<y<1: ", Assuming[assms,Glim[y,w]]];]
(* Examples: *)
(* In[]= *)
FindLimit[ NormalDistribution[\[Mu],\[Sigma]],{\[Mu]\[Element]Reals ,\[Sigma]>0} ]
(* Out[]= *)
0<y<1: 1
(* In[]= *)
FindLimit[ LogNormalDistribution[\[Mu],\[Sigma]],{\[Mu]\[Element]Reals ,\[Sigma]>0} ]
(* Out[]= *)
0<y<1: y
(* In[]= *)
FindLimit[ GammaDistribution[\[Alpha],\[Beta]],{\[Alpha]>0,\[Beta]>0} ]
(* Out[]= *)
0<y<1: (E^(w/\[Beta]) (1-E^(-((w y)/\[Beta]))))/(-1+E^(w/\[Beta]))