Distance of squared random variables and upper bound

Question

I got two sequences of random variables, $(X_n)_n$ and $(Y_n)_n$, and I know that

$| X_n - Y_n | \leq C a_n $, for some constant $C$ or equivalently $|X_n - Y_n | = \mathcal{O}(a_n)$.

Now I want to know what this means for the distance of squared random variables $| X_n^2 - Y_n^2 |$ which I can bound by

$|X_n^2 - Y_n^2 | = | X_n - Y_n | | X_n + Y_n | \leq C a_n ( |X_n - Y_n | + 2|X_n|) = C^2 a_n^2 + 2Ca_n |X_n|$

My problem is now that I dont have a direct bound for $|X_n|$ or $|Y_n|$, only for the difference $|X_n-Y_n|$, so I cannot simplify the last term more from what I see. Does anyone know how to proceed? Thanks!

Answer 1

You cannot do much unless you have some more information. For example if $X_n=n$ and $Y_n=n+\frac 1 {\sqrt n}$ then $|X_n-Y_n|=\frac 1 {\sqrt n}$ so the hypothesis holds with $a_n=\frac 1 {\sqrt n}$ but $|X_n^{2}-Y_n^{2}| $ tends to $\infty$.