Suppose $\{\xi_i\}_{i=1}^\infty$ is a real-valued stochastic process such that $\xi_i$ is $F_{i}$ measurable and $\xi_i$ is conditionally $\sigma$-sub-Gaussian for some $\sigma\geq 0$, which means $$ E[\exp (\lambda \xi_i)|F_{i-1}]\leq \exp (\frac{\lambda^2\sigma^2}{2})~~~\forall \lambda \in R $$
My question is, whether the following is true $$ E[\exp (\xi_j\xi_i)|F_{i-1}]\leq \exp (\frac{\xi_j^2\sigma^2}{2})~~~\forall j<i $$ and $$ E[\exp (\xi_j\xi_i)|F_{i-1}]\leq \exp (\frac{\xi_i^2\sigma^2}{2})~~~\forall j<i $$
The intuition is to consider $\xi_i$ and $\xi_j$ as $\lambda$ in definition. Moreover, how about functions of $\xi_i$ and $\xi_j$. $$ E[\exp (f(\xi_j)\xi_i)|F_{i-1}]\leq \exp (\frac{f(\xi_j)^2\sigma^2}{2})~~~\forall j<i $$ and $$ E[\exp (\xi_jf(\xi_i))|F_{i-1}]\leq \exp (\frac{f(\xi_i)^2\sigma^2}{2})~~~\forall j<i $$