The question:
Let $X_1,....,X_n$ i.i.d. random variables $\sim N(\mu, \sigma^2 )$, and $Y = (X_1-\bar{X}, ..., X_n -\bar{X})$, show that $$M_Y(t) = \exp \left\{ \frac{\sigma^2}{2} \sum_{i=1}^n (t_i - \bar{t})^2\right\},$$ where $t = (t_1,..., t_n), \; \bar{t} = n^{-1} \sum_{i =1}^ n t_i$.
Now, I believe the following could be helpful,
$$M_{Y} (t_1,...,t_n) = \prod_{i = 1} ^n M_{X_i - \bar{X}}(t_i) = \prod_{i = 1} ^n E\left[ e^{t_i X_i - t_i \bar X}\right]$$
I also know that $M_{X_i - \bar X} (t_i) = \exp \left\{\frac{t^2 \sigma^2 }{2}(1 - 1/n)\right\}$. But I can't seem to make any progress, could someone point me in the right direction.
You can develop and apply properties \begin{align*} M_{Y}(t_1,t_2,...,t_n) = \mathbb{E}\left[\exp \left(\sum_{i=1}^n t_i\left(X_i-\overline{X}\right)\right)\right] & = \mathbb{E} \left[ exp \left( \sum_{i}^{n} t_{i}X_{i} - n \overline{t} \sum_{i}^{n} \frac{X_i}{n} \right) \right] \\ & = \mathbb{E} \left[ exp \left( \sum_{i}^{n} t_{i}X_{i} - \overline{t} \sum_{i}^{n} X_i \right) \right] \\ & = \mathbb{E} \left[ exp \left( \sum_{i}^{n}X_i \left( t_{i} - \overline{t} \right) \right) \right] \\ & = \prod_{i = 1}^{n} \mathbb{E} \left[ exp \left( X_i \left( t_{i} - \overline{t} \right) \right) \right] \\ & = \prod_{i = 1}^{n} M_{X_{i}}(t_i - \overline{t}) \end{align*} And I'll leave it here, it's just developing the generator moment functions.