Given is $(X_{t})_{t \in \mathbb{N}}$ a sequence of bounded independent random variables which are not $\mathbb{P}$-a.s. constant. Let $S_{t}=\sum^{t}_{i=1}X_{i}X_{i+1}$ and $\mathcal{F}^{X}_{t}= \sigma(X_{1}, X_{2}, \dots, X_{t})$.
I am trying to figure out when is $S_{t}$ an $\mathcal{(F}^{X}_{t})$-martingale.
According to the definition of a martingale, we need 3 conditions to be satisfied:
1)Every $X_{t}$ is $\mathcal{F}_{t}$-measurable for each $t \in \mathbb{N_{0}}$.
2)$\mathbb{E}[X_{t}]< \infty$ for each $t \in \mathbb{N_{0}}$.
3)$\mathbb{E}^{\mathbb{P}}[X_{t+1} \rvert \mathcal{F}^{X}]=X_{t}$ for every $t \in \mathbb{N_{0}}$.
Can anybody help me with this question, please? Thank you very much in advance!