Evaluate graduate students' mastery of modes of convergence for sequences of random variables: precise definitions of almost sure, in probability, in L^p, and in distribution; prove standard implications (a.s. ⇒ in probability ⇒ in distribution; L^p ⇒ in probability) and demonstrate failure of converses with counterexamples; apply key results (Dominated Convergence, Slutsky, Continuous Mapping, Portmanteau) and basic inequalities (Markov, Chebyshev) in short proofs and interpret consequences for expectations and distributional limits.