Lecture 5 Flashcards
Define convergence in distribution.
Does Convergence in distribution imply convergence of the mean? Give an example.
Under which condition does convergence in distribution imply convergence of the mean?
If the sequence {X^k} is UI, then E(|X|^k) -> E(|X|^k)
State Theorem 17. Hint, it relates convergence in distribution with Convergence of the mean.
State theorem 18.
State theorem 19.
What does the combination of theorem 18 and 19 imply?
What is theorem 20.
Using theorem s you already know, prove that theorem 20 holds for vectors.
Using theorem 17, prove and state the continuous mapping theorem.
State the difference between the Continous mapping theorem, slutzky’s theorem and Cramer’s Theorem.
Cramer: Cramer applies to the product of a random variablesconverging in distributionand another r.v converging in probability.
Slutzky: Applies to functions of r.v that converge in probability.
Continuous Mapping Theorem: Applies to functions of r.v that converge in distribution.
What is theorem 23 and its corollaries?
State and prove theorem 24. Hint: Relationship between convergence in distribution and Op(1).
State theorem 25
