the implications among the modes of convergence

There are four standard ways for random variables to converge — almost surely, in r-th mean, in probability, and in distribution — and a beginner's first question is: which is stronger than which? The reassuring news is that they line up in a clear partial order, with one mode (convergence in probability) as the central hub that everyone passes through on the way to the weakest mode.

The core chain is this. Almost sure convergence implies convergence in probability. Convergence in r-th mean ALSO implies convergence in probability (via Markov's inequality). And convergence in probability implies convergence in distribution. So convergence in distribution is the weakest, sitting at the bottom; convergence in probability sits in the middle as a common consequence of the two strong modes; and the two strong modes — almost sure and r-th mean — are NOT comparable to each other (neither implies the other). The arrows go one way only: none of these implications reverses in general.

Knowing this map saves real work and prevents real errors. To prove the weakest claim you reach for the easiest tool (characteristic functions for distribution); to claim the strongest you must do more (Borel-Cantelli for almost sure). The non-reversals each come with a famous counterexample: a spike of height n with probability 1/n converges in probability but not in mean; the 'typewriter' sequence converges in probability but not almost surely; and convergence in distribution to a constant is special — there it DOES upgrade back to convergence in probability, the one tidy exception worth remembering.