2212 CHAPTER 64. WIENER PROCESSES
The second term clearly converges to 0 as n→ ∞. Consider the first term. To simplify, lett = x−2p|h|
|h| . Then this term reduces to(1
(n+1)!
)2p 22p(n+1) |h|2p(n+1) |h|√2π
e2p|h|2∫R
e−t2t2p(n+1)dt
= 2(
1(n+1)!
)2p 22p(n+1) |h|2p(n+1) |h|√2π
e2p|h|2∫
∞
0e−t2
t2p(n+1)dt
Now let t2 = u. Then this becomes
2(
1(n+1)!
)2p 22p(n+1) |h|2p(n+1) |h|√2π
e2p|h|2∫
∞
0e−uup(n+1)u−(1/2) 1
2du
=
(1
(n+1)!
)2p 22p(n+1) |h|2p(n+1) |h|√2π
e2p|h|2∫
∞
0e−uup+np− 1
2 du
≤ C (h)(2 |h|)2p(n+1) 1(n+1)!
1
((n+1)!)2p−1 Γ
(p(n+1)− 1
2
)= C (h)
(2 |h|)2p(n+1)
(n+1)!Γ(
p(n+1)− 12
)((n+1)!)2p−1
≤ C (h)
(22 |h|2
)p(n+1)
(n+1)!Γ(
p(n+1)− 12
)((n+1)!)2p−1
= C (h)
(22 |h|2
)p(n+1)
(n+1)!(p(n+1))!
((n+1)!)2p−1
this converges to 0 as n→ ∞. This is obvious for (22|h|2)p(n+1)
(n+1)! . Consider (p(n+1))!((n+1)!)2p−1 .By
the ratio test, ∑n(p(n+1))!
((n+1)!)2p−1 < ∞ so this also converges to 0. The details of this ratio testargument are as follows. The ratio, after simplifying is
p factors︷ ︸︸ ︷(pn+2p)(pn+2p−1) · · ·(pn+ p+1)
(n+2)2p−1 ≤ pp (n+ p)p
(n+2)2p−1
which clearly converges to 0 since 2p−1 > p since p is an integer larger than 1.
Therefore,{∣∣∣W (h)n+1
(n+1)!
∣∣∣eW (h)}∞
n=1is bounded in L2p (Ω). Then
∫Ω
∣∣∣∣∣ n
∑k=0
W (h)k
k!− eW (h)
∣∣∣∣∣p
dP→ 0
because the integrand is bounded by(∣∣∣W (h)n+1
(n+1)!
∣∣∣eW (h))p
and it was just shown that these
functions are bounded in L2 (Ω) . Therefore, the claimed convergence follows from theVitali convergence theorem.