1. 4. Convergence of random variables  Convergence in probability  Convergence in distribution  Convergence in quadratic mean  Properties  The law of large numbers  The central limit theorem  Delta method 1

2. CONVERGENCE OF RANDOM VARIABLES Convergence in probability 2

3. Convergence in distribution 3 at any continuity point of CONVERGENCE OF RANDOM VARIABLES

4. Convergence in quadratic mean 4 CONVERGENCE OF RANDOM VARIABLES

5. Properties 5 CONVERGENCE OF RANDOM VARIABLES (i) (ii) (i) (ii) (iii) Then:

6. Properties 6 (iv) (v) Let g() be a continuous function. Then: (i) (ii) CONVERGENCE OF RANDOM VARIABLES

7. The law of large numbers 7 X with ; i. i. d. sample. Let. Then: CONVERGENCE OF RANDOM VARIABLES

8. The central limit theorem 8 X with ; i. i. d. sample. Let. Then: CONVERGENCE OF RANDOM VARIABLES

9. The central limit theorem 9 Remark: CONVERGENCE OF RANDOM VARIABLES (good for n  30)

10. Delta method 10 Suppose that CLT holds: g() differentiable function. Then: i. e., CONVERGENCE OF RANDOM VARIABLES