63.8. LEVY’S THEOREM 2175

and so it also follows ∣∣∣∣E (eiλX(τε∧t j))

eλ22 t j −E

(eiλX(τε∧t j−1)e

λ22 t j−1

)∣∣∣∣≤ O(1)

[δ n−E

(y2

j)+δ no(1)

]Now also remember

y j = X (τε ∧ t j)−X(τε ∧ t j−1

)and that

{X (τε ∧ t j)

}j is a martingale. Therefore it is routine to show,

E(y2

j)= E

(X (τε ∧ t j)

2)−E

(X(τε ∧ t j−1

)2).

and so ∣∣∣∣E (eiλX(τε∧t j))

eλ22 t j −E

(eiλX(τε∧t j−1)e

λ22 t j−1

)∣∣∣∣≤ O(1)

[δ n−

(E(

X (τε ∧ t j)2)−E

(X(τε ∧ t j−1

)2))

+δ no(1)]

and so, summing over all j = 1, · · · ,2n(ε),∣∣∣∣E (eiλX(τε∧b))

eλ22 b−E

(eiλX(a)e

λ22 a)∣∣∣∣

≤ O(1)((1+o(1))(b−a)−

(E(

X (τε ∧b)2)−E

(X (a)2

))). (63.8.48)

Now recall 63.8.42 which saidP([τε < b])< ε.

Let εk ≡ 2−k and then by the Borel Cantelli lemma,

X (τε ∧b)→ X (b)

a.e. since if ω is such that convergence does not take place, ω must be in infinitely many ofthe sets,

[τεk < b

], a set of measure 0. Also since

{X (τε ∧ t j)

}j is a martingale, it follows

from optional sampling theorem that{

X (a)2 ,X (τε ∧b)2 ,X (b)2}

is a submartingale andso ∫

[X(τε∧b)2≥α]X (τε ∧b)2 dP≤

∫[X(τε∧b)2≥α]

X (b)2 dP

and also from the maximal inequalities, Theorem 60.6.4 on Page 1969 it follows

P([

X (τε ∧b)2 ≥ α

])≤ 1

αE(

X (b)2)

and so the functions,{

X(τεk ∧b

)2}

εkare uniformly integrable which implies by the Vitali

convergence theorem, Theorem 11.5.3 on Page 257, that you can pass to the limit as εk→ 0