1974 CHAPTER 60. CONDITIONAL, MARTINGALES

60.7 The Submartingale Convergence TheoremWith this estimate it is now possible to prove the amazing submartingale convergence the-orem.

Theorem 60.7.1 Let {Xn} be a real valued submartingale such that

E (|Xn|)< M

for all n. Then there exists X ∈ L1 (Ω,F ) such that Xn (ω) converges to X (ω) a.e. ω andX ∈ L1 (Ω) .

Proof: Let a < b be two rational numbers. From Theorem 60.6.9 it follows that for allN, ∫

Ω

UN[a,b]dP ≤ 1

b−aE((XN−a)+

)≤ 1

b−a(E (|XN |)+ |a|)≤

M+ |a|b−a

.

Therefore, letting N→∞, it follows that for a.e. ω, there are only finitely many upcrossingsof [a,b] . Denote by S[a,b] the exceptional set. Then letting S ≡ ∪a,b∈QS[a,b], it follows thatP(S) = 0 and for ω /∈ S,{Xn (ω)} is a Cauchy sequence because if

lim supn→∞

Xn (ω)> lim infn→∞

Xn (ω)

then you can pick liminfn→∞ Xn (ω) < a < b < limsupn→∞ Xn (ω) with a,b rational andconclude ω ∈ S[a,b].

Let X (ω) = limn→∞ Xn (ω) if ω /∈ S and let X (ω) = 0 if ω ∈ S. Then it only remainsto verify X ∈ L1 (Ω) . Since X is the pointwise limit of measurable functions, it follows Xis measurable. By Fatou’s lemma,∫

Ω

|X (ω)|dP≤ lim infn→∞

∫Ω

|Xn (ω)|dP

Thus X ∈ L1 (Ω). This proves the theorem.As a simple application, here is an easy proof of a nice theorem about convergence of

sums of independent random variables.

Theorem 60.7.2 Let {Xk} be a sequence of independent real valued random variablessuch that E (|Xk|)< ∞,E (Xk) = 0, and

∞

∑k=1

E(X2

k)< ∞.

Then ∑∞k=1 Xk converges a.e.