4. Convergence of random variables  Convergence in probability  Convergence in distribution  Convergence in quadratic mean  Properties  The law of.

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Presentation transcript:

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

CONVERGENCE OF RANDOM VARIABLES Convergence in probability 2

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

Convergence in quadratic mean 4 CONVERGENCE OF RANDOM VARIABLES

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

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

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

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

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

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