Lecture 3

Question 1

Define the concept of Uniform Integrability.

Answer 1

A sequence {Xi, i ≥ 1} is UI if

lim δ -> inf max i ≥ 1E{|Xi|(i)(|Xi| > δ)} = 0

Question 2

Theorem 9. Give two sufficient conditions for UI.

Answer 2

1. Max E[|Xi|^1+η] < inf. for some η > 0
2. {Xi, i ≥ 1} Identically distributed and E|Xi| < inf.

Question 3

Give a necessary condition for UI. (Theorem 10)

Answer 3

Max E[|Xi|] < inf.

Question 4

State the Weak Law of Large numbers for Indep. UI sequences with mean Zero. (Theorem 11)

Answer 4

Given a sequence of UI independent random variables {Xi} with E[Xi] = 0 for all i, then, the average Xn = (1/n) Σ Xi converges in first mean to 0, and thus converges in probability by theorem 4.

Question 5

How can we make it so that WLLN applies to time series. I.e, that WLLN applies to dependent data.

Answer 5

We can relax the independence assumption to {Xi} being a martingale difference sequence, i.e E(Xi|Xj, j < i) = 0

Question 6

What is the Strong Law of Large Numbers.

Answer 6

Under the same conditions than the WLLN, we can show that the Khinchine’s WLLN almost surely converges to zero, which is stronger than convergence in 1st mean.

Question 7

Is the condition that E[Xi] has mean zero important to the SLLN and WLLN?

Answer 7

No because we can simply demean.

Question 8

Can we relax the condition of identical distribution for Khinchine’s WLLN?

Answer 8

Yes, if the sequence is independent and satisfies the following:
for some δ > 0, Σ E[Xi^(1+δ)] / i^(1+δ) is finite, then
(1/n) Σ(Xi - μi) converges almost surely to zero.

A sufficient condition for the one above is that sup E[Xi^(1+δ)] <= C < inf.

Question 9

What is Khinchine’s WLLN.

Answer 9

If {Xi} is iid with finite first moment and mean zero, the the average of Xi converges in probability.

Question 10

Define the notion of a Generalized Linear Process.

Answer 10

Let {ei, -inf. < i < inf.} be an INDEPENDENT, UI sequence of mx1 random vectors where E[ei] = 0. Let {Aj} be a sequence of mxm non-random matrices and:

ui = Σ Aj*e(i-j) with the Sum of Nomrs of Aj smaller than infinity. Then ui is a generalized Linear Process (GLP). Examples include MA(q), AR(p) or ARMA(p,q) if ei are iid with finite first moments.

Question 11

State theorem 12.

Answer 11

If a data generating process is GLP, then the average of Xi converges in 1st mean to zero.