744 CHAPTER 27. ANALYTICAL CONSIDERATIONS

If we had not replaced Xk with φ (Xk) , it would have been possible for φ (Xk+m (ω)) to beless than a and the zero in the above could have been a negative number.

Therefore from 27.7,

(b−a)E(U[a,b]

)≤ E (φ (Xn)−φ (X1))≤ E (φ (Xn)−a)

= E((Xn−a)+

)≤ |a|+E (|Xn|) ■

With this estimate, the amazing sub-martingale convergence theorem follows. Thisincredible theorem says that a bounded in L1 sub-martingale must converge a.e.

Theorem 27.3.9 (sub-martingale convergence theorem) Let {Xi}∞

i=1 be a sub-mart-ingale with K ≡ sup{E (|Xn|) : n≥ 1}< ∞. Then there exists a random variable X∞, suchthat E (|X∞|)≤ K and limn→∞ Xn (ω) = X∞ (ω) a.e.

Proof: Let a,b ∈ Q and let a < b. Let Un[a,b] (ω) be the number of upcrossings of

{Xi (ω)}ni=1. Then let

U[a,b] (ω)≡ limn→∞

Un[a,b] (ω) = number of upcrossings of {Xi} .

By the upcrossing lemma, E(

Un[a,b]

)≤ E(|Xn|)+|a|

b−a ≤ K+|a|b−a and so by the monotone conver-

gence theorem, E(U[a,b]

)≤ K+|a|

b−a < ∞ which shows U[a,b] (ω) is finite a.e., for all ω /∈ S[a,b]where P

(S[a,b]

)= 0. Define S ≡ ∪

{S[a,b] : a,b ∈Q, a < b

}. Then P(S) = 0 and if ω /∈ S,

{Xk}∞

k=1 has only finitely many upcrossings of every interval having rational endpoints.Thus, for ω /∈ S,

lim supk→∞

Xk (ω) = lim infk→∞

Xk (ω) = limk→∞

Xk (ω)≡ X∞ (ω) .

Letting X∞ (ω) = 0 for ω ∈ S, Fatou’s lemma implies∫Ω

|X∞|dP =∫

Ω

lim infn→∞|Xn|dP≤ lim inf

n→∞

∫Ω

|Xn|dP≤ K ■

27.4 Characteristic Functions and IndependenceThere is a way to tell if random vectors are independent by using their characteristic func-tions.

Proposition 27.4.1 If X i is a random vector having values in Rpi , then the randomvectors are independent if and only if

E(eiP)= n

∏j=1

E(eit j ·X j

)where P≡ ∑

nj=1 t j ·X j for t j ∈ Rp j .

The proof of this proposition will depend on the following lemma.