Suppose I have a stochastic process $\{X_i: i\in\mathbb{N}\}$ on $(\Omega,\mathcal{F},P)$. Assume $E(X_i)=0$ and $E(X_i^4)\leq C$ for some $C>0$ , for any $i$. Since no distribution was assumed, I understand that each random variable can induce its own distribution $P_{X_i}$. So, I cannot conclude, e.g., that $E(X_1X_2X_3X_4)=E(X_1^4)<C$. In order to show that the expectation of any product $X_iX_jX_kX_l$ is also bounded, I have to show that $E(X_iX_jX_kX_l) \leq E(X_m^4)$, for some $m\in\{i,j,k,l\}$.
I will show my proof, and I would apprecite to receive feedbacks.
By $$-1\leq \frac{Cov(X_iX_j,X_kX_l)}{\sqrt{Var(X_iX_j)Var(X_kX_l)}}\leq 1$$ it follows that $\mid Cov(X_iX_j,X_kX_l)\mid\leq \max(Var(X_iX_j),Var(X_kX_l))$. Hence, for some $m_1,m_2\in\{i,l,k,l\}$, \begin{align} &\mid Cov(X_iX_j,X_kX_l)\mid \leq Var(X_{m_1}X_{m_2})\leq E(X_{m_1}^2X_{m_2}^2)\\ =&\mid Cov(X_{m_1}^2,X_{m_2}^2)\mid\leq Var(X_m^2)\leq E(X_m^4)\leq C \end{align} for some $m\in\{m_1,m_2\}$.