3.2 The inhomogeneous setting
In this section we give a companion to Theorem 7 in the inhomogeneous case.
Theorem 9. Set
σ
2
= E[S
2
] σ
2 m
= sup
k ≤n
E[S
2 k
] v =
X
k
E[X
2 k
] + 2E[X
k
S
k −1
]
+
w
p
=
n
X
k=1 k
X
i=1
1
2 kX
k i
2
E[X
k
|F
k i
]k
p
+ X
j i
kE[X
k
X
k j
|H
k, j i
] − E[X
k
X
k j
]X
k, j i
k
p
+ X
j ≤i
|E[X
k
X
k j
]| kX
k i
k
p
.
Then for any a ∈ R
|E[e
iaS
] − e
−a
2
σ
2
2
| ≤ |a|
3
w
1
e
a
2
σ
2 m
−σ
2
2
, 40
E[e
aS
] ≤ exp
v a
2
2 + w
∞
|a|
3
3
. 41
R
EMARK
. In the case of a martingale, σ
2 m
= σ
2
= v and w
p
= P
VarX
k
kX
k
k
p
+ kX
3 k
k
p
2. Proof. Let
λ ∈ C. Set St = S
n −1
+ t X
n
, and ϕt = E[e
λSt
]. The derivative of
ϕ is : ϕ
′
t =λE[X
n
e
λSt
] = λ
2
E[X
n
St] ϕt + wt
42 where, thanks to Lemma 8 with Y = X
n
, Z = X
n 1
, . . . X
n n
−1
, t X
n
, gx = e
λ P
x
i
, |wt| ≤ |λ|
3
sup
Y ∈X
ke
λY
1
+···+Y
n −1
+t Y
n
k
p
w
n q
43 and X is the family of the processes of the form Y
i
= α
i
X
i
where α is any decreasing sequence of
[0, 1]
n
with no more than one term different from 0 or 1, and w
n q
is the term corresponding to k = n in the expression of w
q
. Integrating 42 we get ϕte
−λ
2
R
t
E[X
n
Ss]ds
=ϕ0 + Z
t
e
−λ
2
R
s
E[X
n
Su]du
wsds and since E[X
n
St] is half the derivative of σt
2
= E[St
2
], this rewrites ϕte
−λ
2
σt
2
2
=ϕ0e
−λ
2
σ0
2
2
+ Z
t
e
−λ
2
σs
2
2
wsds. 44
770
If λ = ia ∈ iR, taking p = ∞ in 43, 44 implies
|ϕte
a
2
σt
2
2
− ϕ0e
a
2
σ0
2
2
| ≤ |a|
3
w
n 1
expa
2
sup
≤t≤1
σt
2
2. Now since the function
σt is convex, its supremum over [0, 1] is either σ0 or σ1 hence |ϕ1e
a
2
σ1
2
2
− ϕ0e
a
2
σ0
2
2
| ≤ |a|
3
w
n 1
e
a
2
σ
2 m
2
which implies 40 by induction on n. Now for a fixed real λ ∈ R, let
ϕ
∗
t = sup
Y ∈X
E[e
λY
1
+···+Y
n −1
+t Y
n
]. Equations 43 and 44 with p = 1 imply
ϕt ≤ϕ
∗
0e
λ
2
σt
2
−σ0
2
2
+ |λ|
3
w
n ∞
Z
t
e
λ
2
σt
2
−σs
2
2
ϕ
∗
sds. We have for t
≥ s σt
2
− σs
2
= 2t − sE[X
n
S
n −1
] + t
2
− s
2
E[X
2 n
] ≤ 2t − sE[X
n
S
n −1
]
+
+ t
2
− s
2
E[X
2 n
]. Hence, if we set ut = t E[X
n
S
n −1
]
+
+ t
2
E[X
2 n
]2 ϕt ≤ϕ
∗
0e
λ
2
ut−u0
+ |λ|
3
w
n ∞
Z
t
e
λ
2
ut−us
ϕ
∗
sds. For any Y
∈ X, the same bound holds if X is replaced by Y in the definition of ϕ, since the corre- sponding values of w
n ∞
and ut − us will be smaller either α
n
= 0 and the corresponding value of ut
− us is zero, or Y
n
= α
n
X
n
and Y
i
= X
i
, i n. Hence
ϕ
∗
t ≤ϕ
∗
0e
λ
2
ut−u0
+ |λ|
3
w
n ∞
Z
t
e
λ
2
ut−us
ϕ
∗
sds or
ϕ
∗
te
−λ
2
ut
≤ϕ
∗
0e
−λ
2
u0
+ |λ|
3
w
n ∞
Z
t
e
−λ
2
us
ϕ
∗
sds, and by Gronwall’s Lemma:
ϕ
∗
te
−λ
2
ut
≤ϕ
∗
0e
−λ
2
u0
2
e
t |λ|
3
w
n ∞
which gives for t = 1 ϕ
∗
1 ≤ϕ
∗
0e
λ
2
E[X
n
S
n −1
]
+
+E[X
2 n
]2+|λ|
3
w
n ∞
. This proves 41 by induction on n.
771
3.3 Applications to deviation bounds
Kategori