We are interested in if a sequence of functions converges to . These are easier to contrast with one another if stated in a compact, purely notational form.
Informally, if you pick an , I can find an , such that for all functions after , and never disagree by more than . This is a strong form of convergence, in contrast to the weak
Informally, if you pick an , and a particular I can find an , such that for all functions after , and never disagree by more than on . Though uniform convergence implies pointwise, the converse is not true. To see why, it might be that certain require arbitrarily large . That is, if we think of as a function , we might have that .
Convergence With Probability 1
Convergence in Probability