Let us consider the following sequences: $\{X_{n}\}_{n=1}^{\infty}$ and $\{Y_{n}\}_{n=1}^{\infty}$ such that $X_{n} \xrightarrow{a.s.} a$ and $Y_{n} \xrightarrow{a.s.} a$, for some constant $a > 0$. Next, assume that $$ \sqrt{n}(X_{n} - a) \xrightarrow{d} Z_{x} $$ and $$ \sqrt{n}(Y_{n} - a) \xrightarrow{d} Z_{y}, $$ where $Z_{x}$ and $Z_{y}$ are some random variables.
Next, assume that $X_{n} \leq Y_{n}$ for all $n$.
Can we say something about the sign of sequence $$ \frac{X_{n}}{n-1} - \frac{Y_{n}}{n} $$ when $n\to\infty$?
I believe that the answer is generally "no". First, obviously if $X_n=Y_n$ then the sign is trivially +. We want to show that exist $X,Y$ such that the sign will be always -.
Let $a=1/2$. Let $X_n= a + \frac{1}{n}$ and $Y_n= a + \frac{2}{n}$. You can add some noise if you dont want $Z$ to be degenerate. Then, $\frac{X_n}{n-1} - \frac{Y_n}{n} \approx \frac{a-1}{n(n-1)}<0$ for all $n>2$.