Convergence of random variables What is the difference between convergence in distribution, convergence in probability and almost sure convergence?

References

Convergence of random variables
Convergence in distribution means pointwise convergence of the CDFs.  It's not a form of convergence of the random variables per se. 

Convergence in probability says that for every $\epsilon &gt; 0$, the probability of the sequence of random variables being more than $\epsilon$ from the limiting random variable goes to $0$. Convergence in probability is also called convergence in measure.

Convergence almost everywhere = almost sure convergence = pointwise convergence of the random variables except possibly on a set of measure zero.

Here is a diagram showing which modes of convergence imply which other modes.
A sequence X1  is. 1,1,2,1,1,2 ... 1,1,2...
A sequence X2 is. 2,1,1,2,1,1,... 2,1,1.

Assume a classical setup for the measure.

X1 and X2 converge in distribution.  Since both random variables output the value "1" with 2/3 probability, and "2" has probability 1/3.  

But X1 will never converge in probability to X2 because as the number of samples increases two thirds  of the samples from the random variables will still be different from each other.  Obviously they are not getting close to each other as they should in convergence in probability.