Lecture 4.2

Question 1

Define O(f_n) and o(f_n)

Answer 1

Question 2

Define Op(f_n) and op(f_n)

Answer 2

Question 3

Give a simpler definition of op(1).

Answer 3

Question 4

State two relationships between convergence in probability to a constant and stochastic order of magnitude.

Answer 4

Question 5

State and prove the two relationships between convergence in probability to a constant and stochastic order of magnitude.

Answer 5

Question 6

Provide a more intuitive definition of the difference between Op() and op()

Answer 6

Question 7

State the three results relating different stochastic orders of magnitude.

Answer 7

Question 8

Which is the direction of implication between Op(fn) and op(fn)? Prove it.

Answer 8

Question 9

State and prove the relationship between stochastic orders of magnitude relating fn and gn if fn/gn goes to 0.

Answer 9

Question 10

State and prove the relationship between stochastic order of magnitude and rate of convergence provided a moment exists.

Answer 10

Question 11

State the 4 results relating the stochastic orders of magnitudes of combinations of RVs.

Answer 11

Question 12

State and prove the stochastic order of magnitude of multiplication of two RVs. (for big and little o)

Answer 12

Question 13

State and prove the stochastic order of magnitude of the sum of two RVs. (for big and little o)

Answer 13

Question 14

State and prove the relationship between the multiplication of RVs that are Op(fn) and op(gn).

Answer 14

Question 15

What is the rate of convergence of Beta hat in LSE? Show it.

Answer 15