Convergence of random variables

Question 1

What is the Markov Inequality lemma?

Answer 1

Question 2

Prove the following lemma.

Answer 2

Question 3

Define converges pointwise or surely.

Answer 3

Question 4

Define converges to X in probability.

Answer 4

Question 5

Define converges to X almost surely.

Answer 5

Question 6

Define converges to X in the mean or in L¹.

Answer 6

Question 7

Define converges in L^r.

Answer 7

Question 8

What are the main four types of convergence?

Answer 8

1. In probability
2. Almost surely
3. In the mean/ in L¹
4. In L^r

Question 9

What does the term almost surely mean?

Answer 9

It means that something occurs excpet on an event with probability zero.

Question 10

What is convergence in probability similar to?

Answer 10

Convergence almost surely - except it is weaker

Question 11

What is the sum convergence lemma?

Answer 11

Question 12

Prove the following lemma.

Answer 12

Question 13

What is the lemma about convergence in L^r and convergence in probabiliity?

Answer 13

Question 14

Prove the following lemma.

Answer 14

Question 15

What is the lemma about convergence in probability and convergence almost surely?

Answer 15

Question 16

Prove the following lemma.

Answer 16

Question 17

Does the converse to this lemma apply?

Answer 17

No - convergecne in probability does not imply convergence in L¹ or L^r.

Question 18

Does the converse to this lemma apply?

Answer 18

No - convergence in probability does not imply convergence almost surely

Question 19

What is the weak law of larger numbers theorem?

Answer 19

Question 20

Prove the following theorem.

Answer 20

Question 21

Why is the weak law of large numbers called weak?

Answer 21

Because the convergence is in probability

Question 22

Why is the strong law of large numbers called strong?

Answer 22

The convergence is almost sure

Question 23

What is the Cauchy-Schwarz theorem?

Answer 23

Question 24

Prove the following theorem.

Answer 24

Question 25

What is the strong law of large numbers theorem?

Answer 25

Question 26

Prove the following theorem.

Answer 26

Question 27

What is the Monotone Convergence theorem?

Answer 27

Question 28

What are three important remarks of the following theorem?

Answer 28

Question 29

What is a corollary to the following theorem for nonnegative decreasing sequences?

Answer 29

Question 30

Prove the following corollary.

Answer 30

Question 31

What is the Dominated Convergence theorem?

Answer 31

Question 32

What is the bounded convergence theorem corollary?

Answer 32

Question 33

Prove the following corollary.

Answer 33

Question 34

What is Fatou's Lemma?

Answer 34

Question 35

Prove the following Lemma.

Answer 35