I am new to stochastic process and wondering is there any theorem on "if the expectation of a stochastic process, say $E(X_n)$, converges when $n \to \infty$"?
I mean, given that $E(X_n)$ is monotonous and $E(|X_n|)<\infty$, the martingale convergence theorem will easily apply. But what can we obtain if there is only "$E(X_n)$ is convergent, $E(|X_n|)<\infty$ "?
I don't think there is much useful you can say just based on the convergence of $\mathbb{E}[X_n].$ Note that this is always the case if $(X_n)$ is a sequence of i.i.d. integrable random variables, so any theorem would need to apply to that case as well. And if $(X_n)$ is a martingale, then this is automatically the case, so it certainly couldn't be any stronger than the standard martingale convergence theorem.
Incidentally, you do need more than just "$\sup_{n \in \mathbb N} \mathbb{E}[X_n^+] < \infty$, $\mathbb{E}[X_n]$ is monotone and $\mathbb{E}[|X_n|] < \infty$" to apply the martingale convergence theorem. You need that $X_n$ is a submartingale, i.e. $X_m \le \mathbb{E}[X_n | \mathcal F_m]$ for all $m \le n$. This is not implied by $\mathbb{E}[X_n]$ being monotone. You could consider a sequence $(Z_n)$ of i.i.d. $N(0,1)$ random variables and let $X_n = Z_n + (1-\frac 1n)$. Then $\mathbb{E}[X_n]$ is increasing to $1$, but $X_n$ is not a submartingale.