We consider the CRR model with $\Omega = \{ −1,1 \}^{T}$, $S_t = S_{t−1}(1 + R_t)$, $S_0 \in \mathbb{R}$, $R_t \in \{a,b\}$ with $−1 < a < 0 < b$, $\ P(\{\omega\}) > 0$ $(\forall \omega \in \Omega)$.
If $\mathbb{Q}$ denotes the unique martingale measure, and $t < T$, what is the distribution of the random variable $\frac{S_T}{S_t}$ under $\mathbb{Q}$?
My first approach: since $S_T = S_0\cdot\prod_{k = 1}^{T}(1 + R_k)$, we have
$$\frac{S_T}{S_t} = \frac{S_0\cdot\prod_{k=1}^{T}(1 + R_k)}{S_0\cdot\prod_{k=1}^{t}(1 + R_k)} = \prod_{k=t+1}^{T}(1 + R_k)$$
However, I don't really know what the distribution of a random variable under a martingale is. I thought about computing the conditional expectation $\mathbb{E}_{\mathbb{Q}}[\frac{S_T}{S_t}]$ but since I have no idea what the distribution under a martingale measure is, I couldn't proceed. Is there anyone who could help me with that?