26.7. 0,1 LAWS 729

Theorem 26.7.5 Let {Xk} be a sequence of independent random variables havingvalues in Z a Banach space. That is, the σ algebras σ (Xk) are independent. Then

A≡ {ω : {Xk (ω)} converges}

is a tail event. So is

B≡

{ω :

{∞

∑k=1Xk (ω)

}converges

}.

Proof: Since Z is complete, A is the same as the set where {Xk (ω)} is a Cauchysequence. This set is

∩∞n=1∩∞

p=1∪∞m=p∩l,k≥m {ω : ∥Xk (ω)−X l (ω)∥< 1/n}

Note that

∪∞m=p∩l,k≥m {ω : ∥Xk (ω)−X l (ω)∥< 1/n} ∈ σ

(∪∞

j=pσ (X j))

for every p is the set where ultimately any pair ofXk,X l are closer together than 1/n,

∩∞p=1∪∞

m=p∩l,k≥m {ω : ∥Xk (ω)−X l (ω)∥< 1/n}

is a tail event. The set where {Xk (ω)} is a Cauchy sequence is the intersection of all theseand is therefore, also a tail event.

Now consider B. This set is the same as the set where the partial sums are Cauchysequences. Let Sn ≡ ∑

nk=1Xk. The set where the sum converges is then

∩∞n=1∩∞

p=2∪∞m=p∩l,k≥m {ω : ∥Sk (ω)−Sl (ω)∥< 1/n}

Say k < l and consider for m≥ p

{ω : ∥Sk (ω)−Sl (ω)∥< 1/n, k ≥ m}

This is the same as{ω :

∥∥∥∥∥ l

∑j=k−1

X j (ω)

∥∥∥∥∥< 1/n,k ≥ m

}∈ σ

(∪∞

j=p−1σ (X j))

Thus∪∞

m=p∩l,k≥m {ω : ∥Sk (ω)−Sl (ω)∥< 1/n} ∈ σ(∪∞

j=p−1σ (X j))

and so the intersection for all p of these is a tail event. Then the intersection over all n ofthese tail events is a tail event. ■

From this it can be concluded that if you have a sequence of independent random vari-ables, {Xk} the set where it converges is either of probability 1 or probability 0. A similarconclusion holds for the set where the infinite sum of these random variables converges.This is stated in the next corollary. This incredible assertion is the next corollary.

Corollary 26.7.6 Let {Xk} be a sequence of random variables having values in aBanach space. Then limn→∞Xn (ω) either exists for a.e. ω or the convergence fails totake place for a.e. ω. Also if

A≡

{ω :

∞

∑k=1Xk (ω) converges

},

then P(A) = 0 or 1.