Convergence in probability says , which controls how a single random variable is close to its putative limit . Convergence almost surely controls how the entire tail approaches the limit simultaneously. I.e. we have that for all . This is equivalent to saying that . This means that if you were simulating out the sequence convergence… Continue reading A quick note on convergence in probability vs. convergence a.s.

Exercise: Let be a sequence of independent events with  for all . Show that . Solution: which by independence means that . Now since , we know that for any , which means that Now suppose f.o. Then for sufficiently large . Thus the set f.o., can be written as , which is a countable union of… Continue reading Probability Exercise: If it must happen once it happens infinitely often.