What is the distribution of $\frac{\sum(x_t-a_t)^2}{\sum(x_t-b_t)^2}$

Question

Let $x_t, t \geq 1$, be a sequence of independent random variables, $x_t \sim N(a_t,\sigma^2), t \geq 1$, $a_t, b_t \in \mathbb{R}$. What is the distribution of $S_n$, where: $$S_n=\frac{\sum_{t=1}^n(x_t-a_t)^2}{\sum_{t=1}^n(x_t-b_t)^2}$$