# Detailed definition of the Monotone Convergence Theorem

I have a basic definition of the Monotone Convergence Theorem:

If $\{X_n\}_{n\geqslant 1}$ is a sequence of random variables such that $X_n\nearrow X$ pointwise and $X_n\geqslant 0\ \forall n$, then $E(X_n)\nearrow E(X)$.

My question , how much liberality can made on the convergence part? I mean: which of these (namely - almost sure convergence , convergence in probability, $\&$ convergence in distribution) can be placed in place of the pointwise convergence and still the definition remains exactly same?

It is also true under the convergence in distribution if we replace the assumption of non-decresingness by stochastic dominance, that is, $\left(\mathbb P\left\{X_n\gt t\right\}\right)_{n\geqslant 1}$ is non-increasing for any $t$. Indeed, we have $$\mathbb E\left[X_n\right]=\int_0^{+\infty}\Pr\left\{X_n\gt t\right\}\mathrm dt,$$ $\Pr\left\{X_n\gt t\right\}\to \Pr\left\{X\gt t\right\}$ for $\lambda$-almost every $t$ (by convergence in distribution) and $\Pr\left\{X_n\gt t\right\}\uparrow \Pr\left\{X\gt t\right\}$. The result thus follows by an application of the monotone convergence theorem, which works when the pointwise convergence is replaced by the almost sure convergence.
• You mean if $X_n$ increasingly converges **in distribution** to $X$, then $E(X_n)\nearrow E(X)$ ? Commented Nov 3, 2016 at 11:11
• To be frank, can you please provide me with an example of a sequence $X_n$ increasingly converging to $X$ in distribution? My head has just gone blank right now! I need to do certain things to convince myself.. Commented Nov 4, 2016 at 21:34