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Cramér moderate deviation expansion for martingales with one-sided Sakhanenko’s condition and its
applications
Xiequan Fan, Ion Grama, Quansheng Liu
To cite this version:
Xiequan Fan, Ion Grama, Quansheng Liu. Cramér moderate deviation expansion for martingales with
one-sided Sakhanenko’s condition and its applications. Journal of Theoretical Probability, Springer,
In press. �hal-02487919�
(will be inserted by the editor)
Cram´ er moderate deviation expansion for martingales with one-sided Sakhanenko’s condition and its applications
Xiequan Fan · Ion Grama · Quansheng Liu
Received: date / Accepted: date
Abstract We give a Cram´ er moderate deviation expansion for martingales with differences having finite conditional moments of order 2 + ρ, ρ ∈ (0, 1], and finite one-sided conditional exponential moments. The upper bound of the range of validity and the remainder of our expansion are both optimal. Consequently, it leads to a “half-side” moderate deviation principle for martingales. Moreover, applications to quantile coupling inequality, β-mixing and ψ-mixing sequences are discussed.
Keywords Martingales · Cram´ er moderate deviations · quantile coupling inequality · β-mixing sequences · ψ-mixing sequences
Mathematics Subject Classification (2000) 60G42 · 60F10 · 60E15 · 60F05
1 Introduction
Let (η
i)
i≥1be a sequence of independent and identically distributed (i.i.d.) centered real random variables (r.v.s) satisfying Cram´ er’s condition E exp{c
0|η
1|} < ∞, for some con- stant c
0> 0. Without loss of generality, assume that Eη
21= 1. Cram´ er [6] established an asymptotic expansion of the probabilities of moderate deviations for the partial sums P
ni=1
η
i, based on the powerful technique of conjugate distributions (see also Esscher [10]).
The result of Cram´ er implies that uniformly in 0 ≤ x = o(n
1/2), log P( P
ni=1
η
i> x √ n) 1 − Φ(x) = O
1 + x
3√ n
as n → ∞, (1.1)
where Φ(x) =
√12π
R
x−∞
exp{−t
2/2}dt is the standard normal distribution function. Cram´ er type moderate deviations for sums of independent r.v.s have been obtained by many au- thors. See, for instance, Feller [13], Petrov [18], Sakhanenko [23] and [12]. We refer to the monographs of Petrov [19], Saulis and Statuleviˇ cius [24] and the references therein.
X. Fan (B)
Center for Applied Mathematics, Tianjin University, Tianjin 300072, China E-mail: fanxiequan@hotmail.com
I. Grama·Q. Liu
Univ. Bretagne-Sud, UMR 6205, LMBA, 56000 Vannes, France E-mail: ion.grama@univ-ubs.fr, quansheng.liu@univ-ubs.fr
In this paper we are concerned with Cram´ er moderate deviations for martingales. When the martingale differences are bounded, we refer to Bose [3, 4], Raˇ ckauskas [20–22], Grama and Haeusler [15]. Let (η
i, F
i)
i=0,...,nbe a sequence of square integrable martingale differ- ences defined on a probability space (Ω, F, P), where η
0= 0 and {∅, Ω} = F
0⊆ ... ⊆ F
n⊆ F. Assume that there exist absolute constants H > 0 and N ≥ 0 such that max
i|η
i| ≤ H and
P
ni=1
E[η
2i|F
i−1] − n
≤ N
2. Here and hereafter, the equalities and inequalities between random variables are understood in the P-almost sure sense. From the results in Grama and Haeusler [15], it follows that
log P( P
ni=1
η
i> x √ n) 1 − Φ(x) = O
x
3√ n
, (1.2)
for all √
log n ≤ x = o(n
1/4), n → ∞, and that P ( P
ni=1
η
i> x √ n)
1 − Φ (x) = 1 + o 1
(1.3) uniformly for 0 ≤ x = o n
1/6, n → ∞. In [11] the expansions (1.2) and (1.3) have been extended to the case of martingale differences satisfying the conditional Bernstein condition:
E[η
ik|F
i−1] ≤ 1
2 k!H
k−2E[η
i2|F
i−1] for k ≥ 3 and 1 ≤ i ≤ n, (1.4) where H is a positive absolute constant. We note that the conditional Bernstein condi- tion implies that the martingale differences have finite two-sided conditional exponential moments.
In this paper we extend the expansions (1.2) and (1.3) to the case of martingales with differences having finite (2 + ρ)th moments, ρ ∈ (0, 1], and finite one-sided conditional exponential moments. Assume that there exist constants L, M > 0 and N ≥ 0 such that
E[|η
i|
2+ρe
L ηi+|F
i−1] ≤ M
ρE[η
2i|F
i−1] for all 1 ≤ i ≤ n (1.5) and
n
X
i=1
E[η
i2|F
i−1] − n
≤ N
2. (1.6)
It is easy to see that the conditional Bernstein condition implies (1.5) with ρ = 1, while condition (1.5) generally does not imply the conditional Bernstein condition; see (2.3) for an example. In Theorem 1 of the paper, we prove that if ρ ∈ (0, 1), then for all 0 ≤ x = o n
1/2, log P ( P
ni=1
η
i> x √ n) 1 − Φ (x) = O
1 + x
2+ρn
ρ/2as n → ∞. (1.7)
The expansion (1.7) can be regard as an extension of (1.2). We would like to point out that the range of validity of (1.2) has been enlarged to the classical Cram´ er’s one, and therefore is optimal. Moreover, it is worth mentioning that (1.7) is new even for independent r.v.s.
The last expansion implies that (1.3) holds uniformly in the range 0 ≤ x = o n
ρ/(4+2ρ). We also show that when ρ = 1, equality (1.7) holds for all √
log n ≤ x = o(n
1/2), see Remark 1 for details.
The paper is organized as follows. Our main results for martingales are stated and dis-
cussed in Section 2. Applications to quantile coupling inequality, β-mixing and ψ-mixing
sequences are discussed in Section 3. Proofs of the theorems and their preliminary lemmas
are deferred to Sections 4-10. The proofs of Theorem 1 and Lemma 4 are refinements of Fan et al. [11]. The applications of our results are new, and therefore are of independent interest.
Throughout the paper, c and c
α, probably supplied with some indices, denote respectively a generic positive constant and a generic positive constant depending only on α. Denote by ξ
+= max{ξ, 0} the positive part of ξ.
2 Main results
Let n ≥ 1, and let (ξ
i, F
i)
i=0,...,nbe a sequence of martingale differences, defined on some probability space (Ω, F, P), where ξ
0= 0, {∅, Ω} = F
0⊆ ... ⊆ F
n⊆ F are increasing σ-fields and (ξ
i)
i=1,...,nare allowed to depend on n. Set
X
0= 0, X
k=
k
X
i=1
ξ
i, k = 1, ..., n. (2.1)
Let hX i be the conditional variance of the martingale X = (X
k, F
k)
k=0,...,n: hX i
0= 0, hX i
k=
k
X
i=1
E[ξ
i2|F
i−1], k = 1, ..., n. (2.2) In the sequel we shall use the following conditions:
(A1) There exist a constant ρ ∈ (0, 1] and positive numbers ε
n∈ (0,
12] such that E[|ξ
i|
2+ρe
ε−1n ξ+i|F
i−1] ≤ ε
ρnE[ξ
i2|F
i−1] for all 1 ≤ i ≤ n.
(A2) There exist non-negative numbers δ
n∈ [0,
12] such that |hXi
n− 1| ≤ δ
2na.s.
Condition (A1) can be seen as a one-sided version of Sakhanenko’s condition [23]. In the case of normalized sums of i.i.d. random variables, conditions (A1) and (A2) are satisfied with ε
n= O(
√1n) and δ
n= 0. In the case of martingales, ε
nand δ
nusually are satisfying ε
n, δ
n→ 0 as n → ∞.
Notice that condition (A1) implies that E[e
ε−1n ξi+|F
i−1] must be finite, which means that the positive part of the conditional distribution of ξ
i/ε
nhas an exponential moment, and therefore has conditional moments of any order. However, such an assumption is not required for the negative part of the conditional distribution. For the negative part of ξ
i, we assume a finite conditional moment of order 2+ρ. Thus, condition (A1) does not imply the conditional Cram´ er condition, because E[e
ε−1n |ξi||F
i−1] may not exist.
Let us remark that if ξ
iis bounded, say |ξ
i| ≤ γ
n, then condition (A1) is satisfied with ε
n= e
1/ργ
n. On the other hand, if ξ
isatisfies
ξ
i≤ γ
nand E[|ξ
i|
2+ρ|F
i−1] ≤ τ
nρE[ξ
i2|F
i−1] for all 1 ≤ i ≤ n, (2.3) then condition (A1) is also satisfied with ε
n= max{γ
n, e
1/ρτ
n}. Here we assume that 0 <
γ
n, τ
n≤
12e
−1/ρ.
The following theorem gives a Cram´ er moderate deviation expansion for martingales.
Theorem 1 Assume conditions (A1) and (A2).
[i] If ρ ∈ (0, 1), then there is a constant α > 0, such that for all 0 ≤ x ≤ αε
−1n,
ln P(X
n> x) 1 − Φ (x)
≤ c
α,ρx
2+ρε
ρn+ x
2δ
n2+ (1 + x) (ε
ρn+ δ
n)
. (2.4)
[ii] If ρ = 1, then there is a constant α > 0, such that for all 0 ≤ x ≤ αε
−1n,
ln P(X
n> x) 1 − Φ (x)
≤ c
αx
3ε
n+ x
2δ
n2+ (1 + x) (ε
n| ln ε
n| + δ
n)
. (2.5)
The term ε
n| ln ε
n| in (2.5) cannot be replaced by ε
nunder the stated conditions. Indeed, Bolthausen [2] showed that there exists a sequence of martingale differences satisfying |ξ
i| ≤ 2/ √
n and hX i
n= 1 a.s., such that for all n large enough, sup
x∈R
P(X
n≤ x) − Φ (x)
√ n
log n ≥ c, (2.6)
where c is a positive constant and does not depend on n. If ε
n| ln ε
n| in (2.5) could be improved to ε
n, then we can deduce the following Berry-Esseen bound
sup
x∈R
|P(X
n≤ x) − Φ (x) | ≤ c (ε
n+ δ
n), (2.7) which would violate Bolthausen’s result (2.6). Thus ε
n| ln ε
n| in (2.5) cannot be improved to ε
neven for bounded martingale differences.
If the martingale differences are bounded |ξ
i| ≤ ε
nand satisfy condition (A2), Grama and Haeusler [15] proved the asymptotic expansion (2.5) for all x ∈ [0, α min{ε
−1/2n, δ
−1n}].
Now Theorem 1 holds for a larger range x ∈ [0, αε
−1n] and a much more general class of martingales.
The following corollary states that under conditions (A1) and (A2), the tail probabilities P(X
n> x) can be uniformly approximated by the tail probabilities of the standard normal random variable, when x is in a certain reduced range.
Corollary 1 Assume conditions (A1) and (A2).
[i] If ρ ∈ (0, 1), then for all 0 ≤ x = o(min{ε
−ρ/(2+ρ)n, δ
n−1}),
P(X
n> x) 1 − Φ (x) − 1
≤ c
ρx
2+ρε
ρn+ (1 + x) (ε
ρn+ δ
n)
. (2.8)
[ii] If ρ = 1, then for all 0 ≤ x = o(min{ε
−1/3n, δ
n−1}),
P(X
n> x) 1 − Φ (x) − 1
≤ c
x
3ε
n+ (1 + x) (ε
n| ln ε
n| + δ
n)
. (2.9)
In particular, this implies that
P(X
n> x)
1 − Φ (x) = 1 + o(1)
holds uniformly for 0 ≤ x = o(min{ε
−ρ/(2+ρ)n, δ
−1n}) as max{ε
n, δ
n} → 0.
The inequalities (2.4) and (2.5) together implies that there is a constant α > 0 such that for ρ ∈ (0, 1] and all 0 ≤ x ≤ αε
−1n,
log P(X
n> x) 1 − Φ (x)
≤ c
αx
2+ρε
ρn+ x
2δ
n2+ (1 + x) (ε
ρn| ln ε
n| + δ
n)
. (2.10)
By (2.10), we obtain the following moderate deviation principle (MDP) result.
Corollary 2 Assume that conditions (A1) and (A2) are satisfied with max{δ
n, ε
n} → 0 as n → ∞. Let a
nbe any sequence of real numbers satisfying a
n→ ∞ and a
nε
n→ 0 as n → ∞. Then for each Borel set B ⊂ [0, ∞),
− inf
x∈Bo
x
22 ≤ lim inf
n→∞
1 a
2nlog P
1 a
nX
n∈ B
≤ lim sup
n→∞
1 a
2nlog P
1
a
nX
n∈ B
≤ − inf
x∈B
x
22 , (2.11)
where B
oand B denote the interior and the closure of B respectively.
Since (2.11) may not hold for all Borel set B ⊂ (−∞, 0], inequality (2.11) does not imply the usual MDP, but it can be seen as a “half-side” MDP.
Similar MDP results for martingales can be found in Dembo [8], Gao [14] and Djellout [7]. For the most recent work on MDP for martingales with the conditional Cram´ er condition and the assumption that E[ξ
2i|F
i−1] = 1/n a.s. for all i., we refer to Eichelsbacher and L¨ owe [9] where the authors established a MDP result via Lindeberg’s method.
Remark 1 The sequence of martingale differences (ξ
i, F
i)
i=0,...,ndiscussed so far is stan- dardized. For a general sequence of martingale differences (η
i, F
i)
i≥1, one can restate the conditions (A1) and (A2) as below.
(A1
0) There exist three positive constants ρ ∈ (0, 1], K and L such that E[|η
i|
2+ρe
Kη+i|F
i−1] ≤ L
ρE[η
i2|F
i−1] for all 1 ≤ i ≤ n;
(A2
0) There exists a constant N ≥ 0 such that
n
X
i=1
E[η
2i|F
i−1] − n
≤ N
2a.s.
Under conditions (A1
0) and (A2
0), the inequalities (2.4)-(2.11) remain valid for W
n=
n
X
i=1
η
i√ n (2.12)
instead of X
n, with ε
n= n
−1/2max{K, L} and δ
n= n
−1/2L.
3 Applications
3.1 Quantile coupling inequality
Thanks to the work of Mason and Zhou [17], it is known that Cram´ er moderate deviations can be applied to establishing quantile coupling inequalities. When the martingale differences are bounded, a quantile coupling inequality has been established by Mason and Zhou, see Corollary 2 of [17]. Here, we give a generalization of the inequality of Mason and Zhou [17].
Let (W
n)
n≥1be a sequence of random variables and for each integer n ≥ 1, and let F
n(x) = P(W
n≤ x), x ∈ R,
denote the cumulative distribution function of W
n. Its quantile function is defined by
H
n(s) = inf{x : F
n(x) ≥ s}, s ∈ (0, 1).
Let Z denote a standard normal random variable. Since Φ(Z) =
dU the uniformly distribu- tion random variable, then it is obvious that for each integer n ≥ 1,
H
n(Φ(Z)) =
dW
n,
where =
dstands for equivalent in distribution. For this reason, we define
W
n= H
n(Φ(Z)). (3.1)
By Theorem 1, we prove the following quantile inequality.
Theorem 2 Let (η
i, F
i)
i≥1be a sequence of martingale differences satisfying the following conditional Sakhanenko condition
E[|η
i|
3e
K|ηi||F
i−1] ≤ L E[η
i2|F
i−1], i ≥ 1, and
n
X
i=1
E[η
i2|F
i−1] − n
≤ M a.s.,
where ρ ∈ (0, 1], K, L and M are positive constants. Assume that W
n=
dP
n i=1η
i/ √
n and W
nis defined as in (3.1). There there exist constants α > 0 and D > 0 and an integer n
0such that whenever n ≥ n
0and
|W
n| ≤ α √
n, (3.2)
we have
√ n|W
n− Z|/ ln n ≤ 2D(W
n2+ 1) a.s. (3.3) Furthermore, there exist two positive constants C and λ such that whenever n ≥ n
0, we have for all x ≥ 0,
P √
n|W
n− Z|/ ln n > x
≤ C exp n
− λ x o
. (3.4)
When the martingale differences are bounded, Mason and Zhou [17] proved that (3.3) holds whenever |W
n| ≤ α √
4n. Notice that the bounded martingale differences satisfy the conditional Sakhanenko condition. Moreover, the range |W
n| ≤ α √
4n has been extended to a much larger one |W
n| ≤ α √
n in our theorem.
3.2 β-mixing sequences
Let (η
i)
i≥1be a random process that may be non-stationary. Write S
k,m= P
k+m i=k+1η
i. Assume that there exists a constant ρ ∈ (0, 1) such that
Eη
i= 0 for all i, (3.5)
E|S
k,m|
2+ρ≤ m
1+ρ/2c
2+ρ1, (3.6) and
ES
k,m2≥ c
22m for all k ≥ 0, m ≥ 1. (3.7)
Let F
jand F
j+k∞be σ-fields generated respectively by (η
i)
i≤jand (η
i)
i≥j+k. We say that (η
i)
i≥1is β-mixing if
β(n) =: sup
j
E sup{
P(B|F
j) − P(B)
: B ∈ F
j+n∞} → ∞, n → ∞.
Assume that there exist positive numbers a
1, a
2and τ such that
β (n) ≤ a
1exp{−a
2n
τ}. (3.8)
By Theorem 4.1 of Shao and Yu [25], it is known that (3.6) is implied by the condition that E|η
i|
2+ρ0≤ c
2+ρ1 0for a constant ρ
0> ρ.
Set α ∈ (0,
12). Let m = bn
αc and k = bn/(2m)c be respectively the integers part of n
αand n/(2m). Let
Y
j=
m
X
i=1
η
2m(j−1)+iand S
n=
k
X
j=1
Y
j.
Note that S
nis an interlacing sum of (η
i)
i≥1, and that Var(S
n) = ES
n2.
Theorem 3 Assume conditions (3.5)-(3.8). Suppose that η
i≤ c
3for all i. Then for all 0 ≤ x = o(min{n
12−α, n
α τ /2}),
ln P(S
n/ p
ES
n2> x) 1 − Φ (x)
≤ c
ρ(1 + x)
2+ρn
ρ(12−α). (3.9)
In particular, we have
P(S
n/ p
ES
n2> x)
1 − Φ (x) = 1 + o(1) (3.10)
uniformly for 0 ≤ x = o(min{n
ρ(1−2α)/(4+2ρ), n
α τ /2}).
For a counterpart of Theorem 3 for interlacing self-normalized sums W
n= S
n/ q P
kj=1
Y
j2, we refer to Chen et al. [5].
The following MDP result is a consequence of the last theorem with α = 1/(2 + τ).
Corollary 3 Assume the conditions of Theorem 3. Let a
nbe any sequence of real numbers satisfying a
n→ ∞ and a
nn
−τ /(2τ+4)→ 0 as n → ∞. Then for each Borel set B ⊂ [0, ∞),
− inf
x∈Bo
x
22 ≤ lim inf
n→∞
1 a
2nlog P
1 a
nS
np ES
n2∈ B
≤ lim sup
n→∞
1 a
2nlog P
1 a
nS
np ES
n2∈ B
≤ − inf
x∈B
x
22 ,
where B
oand B denote the interior and the closure of B respectively.
3.3 ψ-mixing sequences
Recall the notations in Section 3.2. We say that (η
i)
i≥1is ψ-mixing if ψ(n) =: sup
j
sup
B
{
P(B|F
j) − P(B)
/P(B) : B ∈ F
j+n∞} → 0, n → ∞. (3.11) Set α ∈ (0,
12). Let m = bn
αc and k = bn/(2m)c be respectively the integers part of n
αand n/(2m), and let
Y
j=
m
X
i=1
η
2m(j−1)+iand S
n=
k
X
j=1
Y
jas in Section 3.2.
Denote
τ
n2= ψ(m) + nψ
2(m) + kψ
1/2(m). (3.12) We have the following Cram´ er moderate deviations for ψ-mixing sequences.
Theorem 4 Assume conditions (3.5)-(3.7) with ρ ∈ (0, 1]. Suppose that η
i≤ c
3for all i, and that τ
n→ 0 as n → ∞.
[i] If ρ ∈ (0, 1), then for all 0 ≤ x = o(n
12−α),
ln P(S
n/ p
ES
n2> x) 1 − Φ (x)
≤ c
ρx
2+ρn
ρ(12−α)+ x
2τ
n2+ (1 + x) 1
n
ρ(12−α)+ τ
n. (3.13) [ii] If ρ = 1, then for all 0 ≤ x = o(n
12−α),
ln P(S
n/ p
ES
n2> x) 1 − Φ (x)
≤ c x
3n
12−α+ x
2τ
n2+ (1 + x) | ln n|
n
12−α+ τ
n. (3.14) In particular, if
ψ(n) = O n
−(2+ρ)(1−α)/α, (3.15)
then
τ
n= O(n
−ρ(12−α)) and P(S
n/ p
ES
n2> x)
1 − Φ (x) = 1 + o(1) (3.16) uniformly for 0 ≤ x = o(n
ρ(1−2α)/(4+2ρ)).
In the independent case, we have ψ(n) = 0 and τ
n= 0. Let α → 0. Then (3.13) and (3.14) recover the optimal range of validity, that is 0 ≤ x = o(n
1/2).
The following MDP result is a consequence of the last theorem.
Corollary 4 Assume the conditions of Theorem 4. Let a
nbe any sequence of real numbers satisfying a
n→ ∞ and a
n/n
12−α→ 0 as n → ∞. Then for each Borel set B ⊂ [0, ∞),
− inf
x∈Bo
x
22 ≤ lim inf
n→∞
1 a
2nlog P
1 a
nS
np ES
n2∈ B
≤ lim sup
n→∞
1 a
2nlog P
1 a
nS
np ES
n2∈ B
≤ − inf
x∈B
x
22 ,
where B
oand B denote the interior and the closure of B respectively.
4 Preliminary lemmas
Assume condition (A1). For any real λ ∈ [0, ε
−1n], define the exponential multiplicative martingale Z(λ) = (Z
k(λ), F
k)
k=0,...,n, where
Z
k(λ) =
k
Y
i=1
e
λξiE[e
λξi|F
i−1] , k = 1, ..., n, Z
0(λ) = 1.
Then for each k = 1, ..., n, the random variable Z
k(λ) defines a probability density on (Ω, F, P). This allows us to introduce the conjugate probability measure P
λon (Ω, F) defined by
dP
λ= Z
n(λ)dP. (4.1)
Denote by E
λthe expectation with respect to P
λ. For all i = 1, . . . , n, let η
i(λ) = ξ
i− b
i(λ) where b
i(λ) = E
λ[ξ
i|F
i−1].
We thus have the following decomposition:
X
k= Y
k(λ) + B
k(λ), k = 1, ..., n, (4.2) where Y (λ) = (Y
k(λ), F
k)
k=1,...,nis the conjugate martingale defined as
Y
k(λ) =
k
X
i=1
η
i(λ), k = 1, ..., n, (4.3)
and B(λ) = (B
k(λ), F
k)
k=1,...,nis the drift process defined as B
k(λ) =
k
X
i=1
b
i(λ), k = 1, ..., n.
In the proofs of theorem, we need a two-sided bound for the drift process B
n(λ). To this end, we prove the following lemma.
Lemma 1 If there exists an s > 2, such that
E[|ξ
i|
se
ε−1n ξ+i|F
i−1] ≤ ε
s−2nE[ξ
i2|F
i−1], (4.4) then
E[ξ
2i|F
i−1] ≤ ε
2n. (4.5)
In particular, condition (A1) implies (4.5).
Proof. By Jensen’s inequality, it is easy to see that
(E[ξ
i2|F
i−1])
s/2≤ E[|ξ
i|
s|F
i−1] ≤ E[|ξ
i|
se
ε−1n ξi+|F
i−1] ≤ ε
s−2nE[ξ
i2|F
i−1].
Thus
(E[ξ
i2|F
i−1])
s/2−1≤ ε
s−2n,
which implies (4.5). u t
Using the last lemma, we establish a two-sided bound for the drift process B
n(λ).
Lemma 2 Assume conditions (A1) and (A2). Then for all 0 ≤ λ ≤ ε
−1n,
|B
n(λ) − λ| ≤ λδ
n2+ c λ
1+ρε
ρn. (4.6)
Proof. By the relation between E and E
λon F
i, we have b
i(λ) = E[ξ
ie
λξi|F
i−1]
E[e
λξi|F
i−1] , i = 1, ..., n.
Jensen’s inequality and E[ξ
i|F
i−1] = 0 imply that E[e
λξi|F
i−1] ≥ 1, λ ≥ 0. Notice that E[ξ
ie
λξi|F
i−1] = E
ξ
i(e
λξi− 1)|F
i−1≥ 0, 0 ≤ λ ≤ ε
−1n. Using Taylor’s expansion for e
x, we get
B
n(λ) ≤
n
X
i=1
E[ξ
ie
λξi|F
i−1]
= λhXi
n+
n
X
i=1
E
ξ
i(e
λξi− 1 − λξ
i) F
i−1.
Recall ρ ∈ (0, 1]. When x ≤ −1, by Taylor’s expansion, it is easy to see that
x(e
x− 1 − x)
≤
x(e
x− 1)
+ x
2≤ 2|x|
2+ρ. When x ∈ (−1, 1), again by Taylor’s expansion, we get
|x(e
x−1 −x)| ≤
12|x|
3e
x+≤ |x|
2+ρe
x+. When x ≥ 1, we have |x(e
x−1− x)| ≤ xe
x≤ x
2+ρe
x. Thus, it holds
|x(e
x− 1 − x)| ≤ 2|x|
2+ρe
x+, x ∈ R. (4.7) By inequality (4.7), we obtain for all 0 ≤ λ ≤ ε
−1n,
B
n(λ) ≤ λhXi
n+ 2λ
1+ρn
X
i=1
E[|ξ
i|
2+ρe
λξ+i|F
i−1]
≤ λhXi
n+ 2λ
1+ρn
X
i=1
E[|ξ
i|
2+ρe
ε−1n ξi+|F
i−1]. (4.8) Condition (A2) implies that hX i
n≤ 2. Combining (4.8), conditions (A1) and (A2) together, we get the upper bound of B
n(λ):
B
n(λ) ≤ λhX i
n+ 2λ
1+ρε
ρnhX i
n≤ λ + λδ
n2+ 4λ
1+ρε
ρn. When x ≤ −1, by Taylor’s expansion, it is easy to see that
e
x−1 − x−
12x
2≤
e
x− 1 −x +
1
2
x
2≤ |x|
2+ρ. When x ∈ (−1, 1), again by Taylor’s expansion, we get
e
x− 1 − x −
12x
2≤
1
6
|x|
3e
x+≤ |x|
2+ρe
x+. When x ≥ 1, we have
e
x− 1 −x−
12x
2≤
e
x−1 −x
+
12x
2≤ x
2+ρe
x. Thus, it holds
e
x− 1 − x − 1 2 x
2≤ |x|
2+ρe
x+, ρ ∈ (0, 1] and x ∈ R. (4.9) Using inequality (4.9), condition (A1) and Lemma 1, we have for all 0 ≤ λ ≤ ε
−1n,
E[e
λξi|F
i−1] = 1 + 1
2 λ
2E[ξ
i2|F
i−1] + E
e
λξi− 1 − λξ
i− 1 2 λ
2ξ
i2F
i−1≤ 1 + 1 2 λ
2E
ξ
i2|F
i−1+ λ
2+ρE
|ξ
i|
2+ρe
ε−1n ξi+F
i−1≤ 1 + 1
2 λ
2+ λ
2+ρε
ρnE
ξ
i2|F
i−1(4.10)
≤ 1 + 2(λε
n)
2. (4.11)
By inequality (4.7) and the fact hX i
n≤ 2, we deduce that for all 0 ≤ λ ≤ ε
−1n,
n
X
i=1
E[ξ
ie
λξi|F
i−1] = λhX i
n+
n
X
i=1
E
ξ
i(e
λξi− 1 − λξ
i) F
i−1≥ λhX i
n− 2λ
1+ρn
X
i=1
E[|ξ
i|
2+ρe
ε−1n ξi+|F
i−1]
≥ λhX i
n− 2λ
1+ρε
ρnhX i
n≥ λ − λδ
n2− 4 λ
1+ρε
ρn.
The last inequality together with (4.11) imply the lower bound of B
n(λ): for all 0 ≤ λ ≤ ε
−1n, B
n(λ) ≥
λ − λδ
2n− 4 λ
1+ρε
ρn1 + 2 (λε
n)
2−1≥ λ − λδ
n2− 6λ
1+ρε
ρn, where the last line follows from the following inequality
λ − λδ
2n− 4 λ
1+ρε
ρn≥ λ − λδ
2n− (6 − 2(λε
n)
2−ρ)λ
1+ρε
ρn≥
λ − λδ
2n− 6λ
1+ρε
ρn1 + 2(λε
n)
2.
The proof of Lemma 2 is finished. u t
Next, we consider the following predictable cumulant process Ψ (λ) = (Ψ
k(λ), F
k)
k=0,...,n: Ψ
k(λ) =
k
X
i=1
log E
e
λξi|F
i−1. (4.12)
The following lemma gives a two-sided bound for the process Ψ (λ).
Lemma 3 Assume conditions (A1) and (A2). Then for all 0 ≤ λ ≤ ε
−1n,
Ψ
n(λ) − λ
22
≤ c λ
2+ρε
ρn+ λ
2δ
n22 . (4.13)
Proof. Using a two-term Taylor’s expansion of log(1 + x), x ≥ 0, we have Ψ
n(λ) − λ
22 hX i
n=
n
X
i=1
E[e
λξi|F
i−1] − 1 − λE[ξ
i|F
i−1] − λ
22 E[ξ
i2|F
i−1]
−
n
X
i=1
1
2 1 + θ
i(E[e
λξi|F
i−1] − 1)
2E[e
λξi|F
i−1] − 1
2, where θ
i∈ (0, 1). Since E[ξ
i|F
i−1] = 0 and E[e
λξi|F
i−1] ≥ 1 for all 0 ≤ λ ≤ ε
−1n, we deduce that for all 0 ≤ λ ≤ ε
−1n,
Ψ
n(λ) − λ
22 hX i
n≤
n
X
i=1
E[e
λξi|F
i−1] − 1 − λE[ξ
i|F
i−1] − λ
22 E[ξ
i2|F
i−1]
+ 1 2
n
X
i=1
E[e
λξi|F
i−1] − 1
2.
Using condition (A1) and the inequalities (4.9)-(4.11), we get for all 0 ≤ λ ≤ ε
−1n,
Ψ
n(λ) − λ
22 hXi
n≤
n
X
i=1
E[e
λξ+i|λξ
i|
2+ρ|F
i−1] + 1 2
n
X
i=1
E[e
λξi|F
i−1] − 1
2≤ λ
2+ρε
ρnn
X
i=1
E[ξ
i2|F
i−1] + (λε
n)
2n
X
i=1
E[e
λξi|F
i−1] − 1
≤ λ
2+ρε
ρnhXi
n+ c
1λ
4ε
2nhX i
n. Thus
Ψ
n(λ) − λ
22 hX i
n≤
1 + c
1(λε
n)
2−ρλ
2+ρε
ρnhXi
n.
Combining the last inequality with condition (A2) and the fact hXi
n≤ 2, we get for all 0 ≤ λ ≤ ε
−1n,
Ψ
n(λ) − λ
22
≤ 2
1 + c
1(λε
n)
2−ρλ
2+ρε
ρn+ λ
2δ
n22 ,
which completes the proof of Lemma 3. u t
In the proof of Theorem 1, we make use of the following lemma, which gives us some rates of convergence in the central limit theorem for the conjugate martingale Y (λ) under the probability measure P
λ. The proof of this lemma is given in Section 10.
Lemma 4 Assume conditions (A1) and (A2).
[i] If ρ ∈ (0, 1), then there is a positive constant α such that for all 0 ≤ λ ≤ α ε
−1n, sup
x
P
λ(Y
n(λ) ≤ x) − Φ(x) ≤ c
α,ρ(λε
n)
ρ+ ε
ρn+ δ
n. In particular, it implies that
sup
x
P(X
n≤ x) − Φ(x) ≤ c
α,ρε
ρn+ δ
n. (4.14)
[ii] If ρ = 1, then there is a positive constant α such that for all 0 ≤ λ ≤ α ε
−1n, sup
x
P
λ(Y
n(λ) ≤ x) − Φ(x) ≤ c
αλε
n+ ε
n| ln ε
n| + δ
n. In particular, it implies that
sup
x
P(X
n≤ x) − Φ(x) ≤ c
αε
n| log ε
n| + δ
n. (4.15)
5 Proof of Theorem 1
Theorem 1 will be deduced by the combination of the following two propositions (1 and
2), which are stated and proved respectively in Subsections 5.1 and 5.2. The proof of the
propositions are similar to the proofs of Theorems 2.1 and 2.2 of Fan et al. [11]. However,
Fan et al. [11] considered the particular case where ρ = 1.
5.1 Proof of upper bound in Theorem 1
The following assertion gives an upper bound for moderate deviation probabilities.
Proposition 1 Assume conditions (A1) and (A2).
[i] If ρ ∈ (0, 1), then there is a constant α > 0 such that for all 0 ≤ x ≤ αε
−1n, P(X
n> x)
1 − Φ (x) ≤ exp
c
α,ρ,1x
2+ρε
ρn+ x
2δ
2n+ (1 + x) (ε
ρn+ δ
n)
. (5.1)
[ii] If ρ = 1, then there is a constant α > 0 such that for all 0 ≤ x ≤ αε
−1n, P(X
n> x)
1 − Φ (x) ≤ exp
c
α,1,1x
3ε
n+ x
2δ
2n+ (1 + x) (ε
n| ln ε
n| + δ
n)
. (5.2) Proof. For all 0 ≤ x < 1, the assertion follows from (4.14) and (4.15). It remains to prove Proposition 1 for all 1 ≤ x ≤ α ε
−1n. Changing the probability measure according to (4.1), we get for all 0 ≤ λ ≤ ε
−1n,
P(X
n> x) = E
λZ
n(λ)
−11
{Xn>x}= E
λexp {−λX
n+ Ψ
n(λ)} 1
{Xn>x}= E
λexp {−λY
n(λ) − λB
n(λ) + Ψ
n(λ)} 1
{Yn(λ)+Bn(λ)>x}. (5.3) Let λ = λ(x) be the positive solution of the following equation
λ + λδ
n2+ cλ
1+ρε
ρn= x, (5.4)
where c is given by inequality (4.6). The definition of λ implies that there exist c
α,0, c
α,1> 0, such that for all 1 ≤ x ≤ α ε
−1n,
c
α,0x ≤ λ ≤ x (5.5)
and
λ = x − c
α,1|θ|(x
1+ρε
ρn+ xδ
2n) ∈ [c
α,0, α ε
−1n]. (5.6) By Lemma 2, it follows that B
n(λ) ≤ x. From (5.3), by Lemma 3 and equality (5.4), we deduce that for all 1 ≤ x ≤ α ε
−1n,
P(X
n> x) ≤ e
cα,2(λ2+ρερn+λ2δn2)−λ2/2E
λe
−λYn(λ)1
{Yn(λ)>0}
. (5.7)
Clearly, it holds E
λe
−λYn(λ)1
{Yn(λ)>0}
= Z
∞0
λe
−λyP
λ(0 < Y
n(λ) ≤ y)dy. (5.8) Similarly, for a standard normal random variable N , we have
E
e
−λN1
{N>0}= Z
∞0
λe
−λyP(0 < N ≤ y)dy. (5.9) From (5.8) and (5.9), it is easy to see that
E
λe
−λYn(λ)1
{Yn(λ)>0}
− E
e
−λN1
{N>0}≤ 2 sup
y
P
λ(Y
n(λ) ≤ y) − Φ(y)
.
Using Lemma 4, we get the following bound: for all 1 ≤ x ≤ α ε
−1n,
E
λe
−λYn(λ)1
{Yn(λ)>0}
− E
e
−λN1
{N>0}≤ c
ρ(λε
n)
ρ+ ε e
n+ δ
n, (5.10)
where
e ε
n=
ε
ρn, if ρ ∈ (0, 1),
ε
n| ln ε
n|, if ρ = 1. (5.11)
From (5.7) and (5.10), we deduce that for all 1 ≤ x ≤ α ε
−1n, P(X
n> x) ≤ e
cα,2(λ2+ρερn+λ2δ2n)−λ2/2E
e
−λN1
{N>0}+ c
ρ(λε
n)
ρ+ e ε
n+ δ
n. Since
e
−λ2/2E
e
−λN1
{N>0}= 1
√ 2π
Z
∞0
e
−(y+λ)2/2dy = 1 − Φ (λ) (5.12) and
1 − Φ (λ) ≥ 1
√ 2π(1 + λ) e
−λ2/2≥ c
α,0√ 2π(1 + c
α,0) 1
λ e
−λ2/2, λ ≥ c
α,0, (5.13) we have the following upper bound for moderate deviation probabilities: for all 1 ≤ x ≤ α ε
−1n,
P(X
n> x)
1 − Φ λ ≤ e
cα,2(λ2+ρερn+λ2δn2)
1 + c
α,ρ,3(λ
1+ρε
ρn+ λ ε e
n+ λδ
n)
. (5.14)
Next, we would like to make a comparison between 1 − Φ(λ) and 1 − Φ(x). By (5.5), (5.6) and (5.13), it follows that
1 ≤ R
∞λ
exp{−t
2/2}dt R
∞x
exp{−t
2/2}dt = 1 + R
xλ
exp{−t
2/2}dt R
∞x
exp{−t
2/2}dt
≤ 1 + c
α,4x(x − λ) exp{(x
2− λ
2)/2}
≤ exp{c
α,5(x
2+ρε
ρn+ x
2δ
2n)}. (5.15) So, it holds
1 − Φ λ
= 1 − Φ(x) exp
|θ
1|c
α,5(x
2+ρε
ρn+ x
2δ
2n) . (5.16) Implementing (5.16) in (5.14) and using (5.5), we obtain for all 1 ≤ x ≤ α ε
−1n,
P(X
n> x)
1 − Φ (x) ≤ exp n
c
α,6(x
2+ρε
ρn+ x
2δ
2n) o
1 + c
α,ρ,7x
1+ρε
ρn+ x ε e
n+ xδ
n≤ exp n c
α,ρ,8x
2+ρε
ρn+ x
2δ
2n+ x ( e ε
n+ δ
n) o .
This completes the proof of Proposition 1. u t
5.2 Proof of lower bound in Theorem 1
The following assertion gives a lower bound for moderate deviation probabilities.
Proposition 2 Assume conditions (A1) and (A2).
[i] If ρ ∈ (0, 1), then there is a constant α > 0 such that for all 0 ≤ x ≤ αε
−1n, P(X
n> x)
1 − Φ (x) ≥ exp
− c
α,ρ,2x
2+ρε
ρn+ x
2δ
n2+ (1 + x) (ε
ρn+ δ
n)
. (5.17)
[ii] If ρ = 1, then there is a constant α > 0 such that for all 0 ≤ x ≤ αε
−1n, P(X
n> x)
1 − Φ (x) ≥ exp
− c
α,1,2x
3ε
n+ x
2δ
2n+ (1 + x) (ε
n| ln ε
n| + δ
n)
. (5.18) Proof. For all 0 ≤ x < 1, the assertion follows from (4.14) and (4.15). It remains to prove Proposition 2 for all 1 ≤ x ≤ αε
−1n, where α > 0 is a small constant. Let λ = λ(x) be the smallest positive solution of the following equation
λ − λδ
n2− cλ
1+ρε
ρn= x, (5.19) where c is given by inequality (4.6). The definition of λ implies that for all 1 ≤ x ≤ αε
−1n,
x ≤ λ ≤ c
α,1x (5.20)
and
λ = x + c
α,2|θ|(x
1+ρε
ρn+ xδ
n2) ∈ [1, ε
−1n]. (5.21) From (5.3), using Lemmas 2, 3 and equality (5.19), we have for all 1 ≤ x ≤ αε
−1n,
P(X
n> x) ≥ e
−c1(λ2+ρερn+λ2δ2n)−λ2/2E
λh
e
−λYn(λ)1
{Yn(λ)>0}i
. (5.22)
In the subsequent we distinguish λ into two cases. First, let 1 ≤ λ ≤ α
1min{ε
−ρ/(1+ρ)n, δ
n−1}, where α
1> 0 is a small positive constant whose exact value will be given later. Note that inequality (5.10) can be established with λ replaced by λ, which, in turn, implies that
P(X
n> x) ≥ e
−c1(λ2+ρερn+λ2δn2)−λ2/2E
e
−λN1
{N>0}− c
ρ,2(λε
n)
ρ+ ε e
n+ δ
n, where e ε
nis defined by (5.11). By (5.12) and (5.13), we get the following lower bound on tail probabilities:
P(X
n> x)
1 − Φ (λ) ≥ e
−c1(λ2+ρερn+λ2δn2)1 − c
ρ,2λ
1+ρε
ρn+ λ e ε
n+ λδ
2n. (5.23) Taking α
1= min{
(8c 1ρ,2)1/(1+ρ)
,
8c1ρ,2
}, we have for all 1 ≤ λ ≤ α
1min{ε
−ρ/(1+ρ)n, δ
n−1}, 1 − c
ρ,2λ
1+ρε
ρn+ λ ε e
n+ λδ
n≥ exp
−2c
ρ,2λ
1+ρε
ρn+ λ e ε
n+ λδ
n. (5.24) Implementing (5.24) in (5.23), we get
P(X
n> x) 1 − Φ (λ) ≥ exp
− c
ρ,3λ
2+ρε
ρn+ λ e ε
n+ λδ
n+ λ
2δ
n2(5.25)
which holds for all 1 ≤ λ ≤ α
1min{ε
−ρ/(1+ρ)n, δ
−1n}.
Next, consider the case of α
1min{ε
−ρ/(1+ρ)n, δ
−1n} ≤ λ ≤ αε
−1n. Let K ≥ 1 be a constant, whose exact value will be chosen later. It is obvious that
E
λh
e
−λYn(λ)1
{Yn(λ)>0}i
≥ E
λh
e
−λYn(λ)1
{0<Yn(λ)≤Kτ}i
≥ e
−λKτP
λ0 < Y
n(λ) ≤ Kτ
, (5.26)
where τ = (λε
n)
ρ+ e ε
n+ δ
n. From Lemma 4, we get P
λ0 < Y
n(λ) ≤ Kτ
≥ P
0 < N ≤ Kτ
− c
ρ,5τ
≥ Kτ e
−K2τ2/2− c
ρ,5τ
≥
Ke
−8K2α− c
ρ,5τ.
Taking α = 1/(16K
2), we obtain P
λ0 < Y
n(λ) ≤ Kτ
≥ 1
2 K − c
ρ,5τ.
Letting K ≥ 8c
ρ,5, we deduce that P
λ0 < Y
n(λ) ≤ Kτ
≥ 3
8 Kτ ≥ 3
8 K max
λ
1+ρε
ρn, λδ
nλ .
Choosing K = max n
8c
ρ,5,
16α−1−ρ 1
3√ π
o
and taking into account that α
1min{ε
−ρ/(1+ρ)n, δ
n−1} ≤ λ ≤ αε
−1n, we get
P
λ0 < Y
n(λ) ≤ Kτ
≥ 2
√ πλ . Since the inequality
√ 2
πλ e
−λ2/2≥ 1 − Φ (λ)
is valid for all λ > 0, it follows that for all α
1min{ε
−ρ/(1+ρ)n, δ
n−1} ≤ λ ≤ αε
−1n, P
λ0 < Y
n(λ) ≤ Kτ
≥
1 − Φ (λ)
e
λ2/2. (5.27)
From (5.22), (5.26) and (5.27), we get P(X
n> x)
1 − Φ (λ) ≥ exp
− c
α,6λ
2+ρε
ρn+ λ e ε
n+ λδ
n+ λ
2δ
2n(5.28) which holds for all α
1min{ε
−ρ/(1+ρ)n, δ
−1n} ≤ λ ≤ αε
−1n.
Combining (5.25) and (5.28) together, we obtain for all 1 ≤ λ ≤ αε
−1n, P(X
n> x)
1 − Φ (λ) ≥ exp
− c
α,ρ,7λ
2+ρε
ρn+ λ ε e
n+ λδ
n+ λ
2δ
2n. (5.29)
By a similar argument as in (5.15), it is easy to see that 1 − Φ (λ) =
1 − Φ(x) exp
−|θ|c
3(x
2+ρε
ρn+ x
2δ
2n) . (5.30) Combining (5.20), (5.29) and (5.30) together, we find that for all 1 ≤ x ≤ αε
−1n,
P(X
n> x) 1 − Φ (x) ≥ exp
− c
α,ρ,8x
2+ρε
ρn+ x ε e
n+ xδ
n+ x
2δ
2n, (5.31)
which gives the conclusion of Proposition 2. u t
6 Proof of Corollary 2
To prove Corollary 2, we need the following two-sides bound on tail probabilities of the standard normal random variable:
√ 1
2π(1 + x) e
−x2/2≤ 1 − Φ(x) ≤ 1
√ π(1 + x) e
−x2/2, x ≥ 0. (6.1) First, we prove that for any given Borel set B ⊂ [0, ∞),
lim sup
n→∞
1 a
2nlog P
1 a
nX
n∈ B
≤ − inf
x∈B
x
22 . (6.2)
Let x
0= inf
x∈Bx. Then it is obvious that x
0≥ 0 and x
0≥ inf
x∈Bx. By Theorem 1, we deduce that
P 1
a
nX
n∈ B
≤ P (X
n≥ a
nx
0)
≤
1 − Φ (a
nx
0) exp
c
α(a
nx
0)
2+ρε
ρn+ (a
nx
0)
2δ
2n+ (1 + (a
nx
0)) (ε
ρn| ln ε
n| + δ
n) . Using (6.1) and the assumption a
nn→ 0, we have
lim sup
n→∞
1 a
2nlog P
1 a
nX
n∈ B
≤ − x
202 ≤ − inf
x∈B
x
22 , which gives (6.2).
Next, we prove that for any given Borel set B ⊂ [0, ∞), lim inf
n→∞
1 a
2nlog P
1
a
nX
n∈ B
≥ − inf
x∈Bo
x
22 . (6.3)
For any ε
1> 0, there exists an x
0∈ B
o, such that x
202 ≤ inf
x∈Bo
