Main Question: Let $\{X_n, n=0,1,2...\}$ be a stochastic process. How are following two statements different?
\begin{align} E[|X_n|]<\infty \text{ for all } n \end{align}
and
\begin{align} \text{ for some } M<\infty, E[|X_n|]\leq M\text{ for all } n \end{align}
A bit more background and my idea: If $X_n$ is a Martingale, we know the first statement holds. However, Martingale Convergence Theorem requires second statement in addition to $X_n$ being a Martingale. Therefore, it seems like two statements are not equivalent but I cannot see the difference.
In the first statement, we do not force an finite upper bound that holds for every $n$. Then, for every $n$ there is an $M_n<\infty$ such that $E[|X_n|]\leq M_n$ and for every $M_n$, there is a $m\neq n$ such that $E[|X_m|]> M_n$. So, for every $n$, there is a finite bound but there is not a single finite value that is true for all $n$. I am not sure if my interpretation is correct. Like I said, I cannot see the difference between two statements and that's the explanation I came up with but I am not convinced.
If someone can help me to understand this, I would be grateful. Thanks.
The distinction exists even without any randomness. Let $x_n = n$. Then $|x_n| < \infty$ for all $n$, but there is no $M < \infty$ such that $|x_n| \le M$ for all $n$.