Chebyshev’s Inequality Markov’s Inequality Proposition 2.1.

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

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,

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

Almost Sure Convergence

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

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

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

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

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