What is the limiting distribution of $R$? For three independent random variable sequences $X_n,Y_n,Z_n$, we have $X_n\xrightarrow{d} N(0,1)$ (which means $X_n$ converges to Standard Gaussian distribution in distribution), $Y_n\xrightarrow{d} N(0,1), Z_n\xrightarrow{d} N(0,1)$. Therefore, the square each of them (say, $X_n^2$) converges to $\chi_1^2$ in distribution.
 A: I believe you need some additional assumptions. Remember convergence in distribution is simply a statement about pointwise convergence of CDFs, so it doesn't tell us about the underlying probability spaces.
First, if $U_n,V_n$ are independent and if $U,V$ are independent and if $U_n\overset{d}{\rightarrow}U,V_n\overset{d}{\rightarrow}V$ then $U_n+V_n\overset{d}{\rightarrow}U+V$.  You can show this using characteristic functions (remember convergence in distribution is equivalent to pointwise convergence in characteristic functions via Levy's continuity theorem).
To appeal to this result, I will assume $X_n,Y_n,Z_n$ are independent and they respectively converge in distribution to $X,Y,Z$ that are also independent and identically standard normal; then I think we can obtain the limiting distribution of $R_n$.
The $O_p(n^{-1/2})$ term converges in distribution to constant zero, so by the continuous mapping theorem and the above independence result, we have
$$R_n\overset{d}{\rightarrow}X^2+Y^2-Z^2=W^2-Z^2$$
where $W^2\sim \chi_2^2,Z^2\sim \chi^2_1 $ are independent. Independence tells us the characteristic function of $W^2-Z^2$ is given by
$$\varphi(t)=(1-2it)^{-1}(1+2it)^{-1/2}.$$
