Chapter 11: Convergence of Random Variables

Learning Objectives

• Distinguish the four modes of convergence (a.s., in probability, in $L^r$, in distribution) and know which implications hold
• State and prove the Weak Law of Large Numbers via Chebyshev's inequality
• State the Strong Law of Large Numbers and prove the finite-fourth-moment case via Borel-Cantelli
• Prove the Central Limit Theorem via characteristic functions and state the Berry-Esseen bound
• Apply the multivariate CLT, the delta method, and Slutsky's theorem
• Explain why the Gaussian distribution appears so pervasively in communications