1. Chebyshev’s Inequality Markov’s Inequality Proposition 2.1.

2. Chebyshev’s Inequality Chebyshev’s Inequality: Proposition 2.2. Consider Example 2a

3. Convergence in probability A sequence of random variables, X 1, X 2, …, converges in probability to a random variable X if, for every  > 0, or equivalently,

4. The weak law of large numbers Theorem 2.1. The weak law of large numbers Proof:

5. Almost Sure Convergence

6. The Strong Law of Large Numbers Theorem 4.1, p. 400

7. Convergence in distribution A sequence of random variables, X 1, X 2, …, converges in distribution to a random variable X if at all points x where F X (x) is continuous. This really says that the CDFs converge

8. Central Limit Theorem Theorem 3.1. For iid random variables X i Consider Examples 3b and 3c, p. 396

9. Central limit theorem for independent random variables Theorem 3.2, p. 399. (a)The is uniformly bounded, meaning for some M, (b) and

10. Jensen’s ineqality Proposition 5.3, p. 409 If f is convex Consider Example 5f.