The difference between convergence in probability and convergence in distribution. I'm confused with the concepts of convergence in probability and convergence in distribution. After reading some examples in Wiki, can I say convergence in probability means the decrease of variance as n goes to infinity, that is we become more and more confident that one outcome in the sample space will happen? Convergence in distribution just indicates the probability distribution.
 A: One of the key differences is that convergence in probability tells you something about random variables being close in a pointwise sense (with a high probability) whereas convergence in distribution says only something about the closeness of the distributions. 
Let me illustrate this with an example: Take to identically distributed random variables $X$ and $Y$ and define $$X_n := Y \qquad \text{for all $n \geq 1$}.$$ Since all the random variables have the same distribution, we clearly have $X_n \to X$ in distribution. On the other hand, we can, in general not, expect that $X$ is close to $Y$ in a pointwise sense. For instance, if $X \sim N(0,1)$ and we set $Y=X_n:=-X$, then $$\mathbb{P}(|X-X_n|>\delta) = \mathbb{P}(|X|>\delta/2)>0$$ which means that $X_n$ does not convergence in probability to $X$.
Convergence in distribution does not even require that the random variables are all defined on the same probability space, i.e. each random variable $X_n$ may be defined on some probability space $(\Omega_n,\mathcal{A}_n,\mathbb{P}_n)$. In particular, it doesn't even make sense to ask whether the random variables $X_n$ are being close to each other since we can't calculate the probability of the set
$$\{\omega \in ??; |X_n(\omega)-X_m(\omega)| >\delta\}$$
(probability with respect to which measure? $\mathbb{P}_n$ or $\mathbb{P}_m$...? In fact, we can't even write down the set properly since the $\omega$'s are elements in different probability spaces.)
