Let $X_1, ..., X_n$ be a sample from $N(\mu, 1)$. Fix $1 \leq m<n$ and define $$T_i= \frac{1}{m}\sum\limits_{j=i}^{i+m-1} X_j,$$ for $i \in \lbrace 1, ..., n-m+1 \rbrace$. We have the test that rejects $H_0 : \mu = 0$ when max$\lbrace T_i \rbrace> c_{\alpha}$. Find $c_\alpha$ in terms of the standard normal distribution.
If the $T_i$ were independent, then they would be i.i.d Normal and the distribution of their maximum is known. But I dont know how to proceed since the $T_i$ are not independent.
