792 CHAPTER 29. MARTINGALES

the last from the monotone convergence theorem. Hence from the above,

E

(∑k|Yk|2

)= ∑

kE(|Yk|2

)≤∑

k∑n

E((Xn,ek)

2H

)= ∑

nE

(∑k(Xn,ek)

2

)= ∑

nE(|Xn|2H

)< ∞

the last by assumption. Therefore, for ω off a set of measure zero, and for Yk (ω) ≡∑

∞n=1 (Xn (ω) ,ek)H which exists a.e. by Theorem 29.3.8, ∑k |Yk (ω)|2 < ∞ and so for a.e.

ω, S (ω)≡ ∑∞k=1 Yk (ω)ek makes sense. Thus for these ω

S (ω) = ∑l(S (ω) ,el)el = ∑

lYl (ω)el ≡∑

l∑n(Xn (ω) ,el)H el

= ∑n

∑l(Xn (ω) ,el)el = ∑

nXn (ω) .■

Now with this theorem, here is a strong law of large numbers.

Theorem 29.3.11 Suppose {Xk} are independent random variables and

E (|Xk|)< ∞

for each k and E (Xk) =mk. Suppose also

∞

∑j=1

1j2 E

(∣∣X j−m j∣∣2)< ∞. (29.9)

Then limn→∞1n ∑

nj=1 (X j−m j) = 0 a.e.

Proof: Consider the sum∞

∑j=1

X j−m j

j.

This sum converges a.e. because of 29.9 and Theorem 29.3.10 applied to the random vec-tors

{X j−m j

j

}. Therefore, from Lemma 26.8.4 it follows that for a.e. ω ,

limn→∞

1n

n

∑j=1

(X j (ω)−m j) = 0.■

The next corollary is often called the strong law of large numbers. It follows immedi-ately from the above theorem.

Corollary 29.3.12 Suppose{X j}∞

j=1 are independent random vectors, λX i = λX j

for all i, j having meanm and variance equal to

σ2 ≡

∫Ω

∣∣X j−m∣∣2 dP < ∞.

Then for a.e. ω ∈Ω, limn→∞1n ∑

nj=1X j (ω) =m