As the title, given $X_1, X_2, \cdots$ and $Y_1, Y_2, \cdots$ where $X_i$ are i.i.d and so are $Y_j$, and $X_i$ and $Y_j$ are independent from each other. I need to find a test statistic $\tau$ and a critical value $c$ s.t. under the null hypothesis $H_0: E(X_1)=E(Y_1)=0$ the rejection probability, i.e., the probability of $\tau > c$ converges to $0.05$.
I have already found a consistent estimator which is $$\hat{\theta}=\frac{1}{n}\left(\sum_{i=1}^nX_i^2+\sum_{i=1}^nY_i^2\right)$$ and I think $$\tau= \frac{\sum_{i=1}^nX_i^2+\sum_{i=1}^nY_i^2}{n\hat{\sigma}(X_i+Y_i)}$$ is a proper test statistic but I am not sure how to show this converges to the desired result. Thank you.