Let's consider sequence of an independent random variables $X_n$ which has the same distribution and we know that for each $n$, $EX_n=0$ the problem is to show that $E\left|X_0X_1+X_1X_2+...+X_{n-1}X_n\right|< \infty$ for $n \ge 1$
My try:
$E|X_0X_1+X_1X_2+...+X_{n-1}X_n|\le E|X_0X_1|+...+E|X_{n-1}X_n|=n \cdot E|X_0X_1| = n\cdot EX_0^2$ but now I don't know how to show this is finite.
If $\left(X_n\right)_{n\geqslant 1}$ is an independent sequence, so is the sequence $\left(\left|X_n\right|\right)_{n\geqslant 1}$. Since $$\left|\sum_{j=0}^{n-1}X_jX_{j+1} \right|\leqslant \sum_{j=0}^{n-1}\left|X_j\right|\left|X_{j+1} \right|$$ and $\left|X_j\right|$ is independent of $\left|X_{j+1} \right|$ for any $j\in\left\{0,\dots,n-1\right\}$, we derive that $$\mathbb E\left|\sum_{j=0}^{n-1}X_jX_{j+1} \right|\leqslant \sum_{j=0}^{n-1}\mathbb E\left|X_j\right|\cdot \mathbb E\left|X_{j +1} \right|, $$ and the right hand side consists of finitely many finite terms.
Notice that we only need pairwise independence and the fact that all the $\left|X_n\right|$ have a finite expectation. In particular, the $X_n$ do not need to be centered.