Convergence Of pGFs

Question 1

What is G(x+y)[theta]

Answer 1

G(x+y) = G(X)[theta] * G(y)[theta]

Question 2

What does a PGF determine and what are P(Z=k) and P(Z=inf)

Answer 2

PGF of an RV Z determines distribution of Z, P(Z=k) = G(Z)^k(0)/k!
P(Z=inf8 = 1- G(Z)(1-)

Question 3

If have RVs X1, X2,… , what are equivalent

Answer 3

If have RVs X1,X2…, following are equivalent:
PGF GXn converges pointwise to Gx as n tends to inf
Distribution of Xn converges to distribution of X, P(Xn=k) = P(X=k) as n tends to inf

Question 4

What is Gk(theta) and G-k(theta)

Answer 4

Gk(theta) is (G1(theta))^k
G-k(theta) = (G-1(theta))^k

Question 5

What is Gx(theta)

Answer 5

Gx(theta) = E[theta^Hx] = sum k>=0 theta^k P(X=k)

Question 6

What is Gx(1-)

Answer 6

Gx(1-) = lim theta tends to 1 from below gx(theta) = P(X < inf)

Question 7

What is Hr

Answer 7

Hr is return time to r Hr = inf(k > 0 : Wk = r)

Question 8

What is Gr

Answer 8

Gr is PGF of Hr given W0 = 0, Gr(theta) = E0(theta^Hr) = E(theta^Hr|W0 = 0)

Question 9

Revise proof of G1(theta)

Answer 9

Revise proof of G1(theta) don’t need to memorise but proof of theorem 6.1

Question 10

What is P0(H0 = inf) and P0(H0 = 2n) = 1/(2n-1) (2nCn) p^n*q^n

Answer 10

P0(H0 = inf) = P0(no return) = mod(p - q)

Question 11

What is E0(H0|H0 < inf) and E0(H0)

Answer 11

E0(H0|H0 < inf) = 1 + mod(p - q)/mod(p - q), E0(H0) = inf, if p,q = 0.5, P0(H0 < inf) = 1