I was unsuccessful in deriving a good estimate of the distance below.
Let $(X_{n})_{n \geqslant 1}$ be a sequence of i.i.d. random variables, and let $(\varepsilon_{n})_{n\geqslant 1}$ be a sequence of independent Rademacher random variables independent of $(X_{n})_{n\geqslant 1}$. For $n\geqslant 1$ and $x>0$, I am interested in a good estimate of $$|P(|X_1+\dots+X_{n}|/V_{n}\geqslant x)-P(|ε_1X_1+...+ε_{n}X_{n}|/V_{n}|\geqslant x)|,$$ where $V_{n}^2=X_1^2+\dots+X_{n}^2$ (we put $0/0=0$).
Thank you,
Aurel