DSpace at VNU: L (1) bounds for some martingale central limit theorems

Lithuanian Mathematical Journal, Vol. 54, No. 1, January, 2014, pp. 48–60

L1 bounds for some martingale central limit theorems
Le Van Dung a,1 , Ta Cong Son b,2 , and Nguyen Duy Tien b,1
a

Faculty of Mathematics, Da Nang University of Education, 459 Ton Duc Thang, Da Nang, Viet Nam
b
Faculty of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Hanoi, Viet Nam
(e-mail: ; ; )

Received September 24, 2013; revised January 7, 2014

Abstract. The aim of this paper is to extend the results in [E. Bolthausen, Exact convergence rates in some martingale
central limit theorems, Ann. Probab., 10(3):672–688, 1982] and [J.C. Mourrat, On the rate of convergence in the martingale central limit theorem, Bernoulli, 19(2):633–645, 2013] to the L1 -distance between distributions of normalized
partial sums for martingale-difference sequences and the standard normal distribution.
MSC: 60F05, 60G42
Keywords: mean central limit theorems, rates of convergence, martingale

1

Introduction and statements of results

2
Let X1 , X2 , . . . , Xn be a sequence of real-valued random variables with mean
√ zero and finite variance σ . Put
S := X1 +X2 +· · ·+Xn . Denote by Fn the distribution functions of S/σ n, and let Φ be the standard normal
distribution function. The classical central limit theorem confirms that if X1 , X2 , . . . , Xn are independent and
identically distributed, then Fn (x) converges to Φ(x) as n → ∞ for all x ∈ R. In 1954, Agnew [1] showed that
the convergence also holds in Lp for p > 1/2. The convergence in the case of p = 1 is called the mean central
limit theorem. The rate of convergence in the mean central limit theorem was also studied by Esseen [7], who

showed that

Fn − Φ

1

= O n−1/2

as n → ∞.

Recently, Sunklodas [12, 13] has extended this result to independent nonidentically distributed random
variables and ϕ-mixing random variables by using the Bentkus approach [2].
Let X = (X1 , . . . , Xn ) be a square-integrable martingale-difference sequence of real-valued random
variables with respect to the σ -fields Fj = σ(X1 , . . . , Xj−1 ), j = 2, 3, . . . , n + 1; F1 = {∅, Ω}. Let
Mn denote the class of all such sequences of length n. If X ∈ Mn , we write σj2 = E(Xj2 | Fj−1 ),
σ 2j = E(Xj2 ), S = S(X) = nj=1 Xj , s2 = s2 (X) = nj=1 σ 2j , V 2 = V 2 (X) = nj=1 σj2 /s2 (X), and
1
2

The research of the author has been partially supported by the Viet Nam National Foundation for Science and Technology Development (NAFOSTED), grant No. 101.03-2012.17.
The research of the author has been partially supported by project TN-13-01.

0363-1672/14/5401-0048 c 2014 Springer Science+Business Media New York

48


L1 bounds for some martingale CLT

49

X p = max1 j n Xj p for 1 p ∞. We denote by N a standard normal random variable; the distribution function and the density function of N are denoted by Φ(x) and ϕ(x), respectively.
If X ∈ Mn , V (X) → 1 in probability, and some Lindeberg-type condition is satisfied, then
lim P

n→∞

S(X)
s(X)

x

= Φ(x)

for all x ∈ R.

For bounds of the convergence rate in this central limit theorem, the following results were shown by
Bolthausen [4].
Theorem 1. (See [4].) Let 0 < α β < ∞, 0 < γ < ∞. There exists a constant 0 < Cα,β,γ < ∞ such that,
for any n 2 and any X ∈ Mn satisfying σj2 = σ 2j a.s., α σ 2j β for 1 j n, and X 3 γ ,
sup P
x∈R

S
s

x − Φ(x)

Cα,β,γ n−1/4 .

Theorem 2. (See [4].) Let γ ∈ (0; +∞). There exists a constant 0 < Cγ < ∞, depending only on γ , such
that, for any n 2 and any X ∈ Mn satisfying X ∞ γ and V (X) = 1 a.s.,
sup P
x∈R

S
s

x − Φ(x)

Cγ

n log n
.
s3

Relaxing the condition that V 2 = 1 a.s., Bolthausen [4] also showed the following result.
Corollary 1. (See [4].) Let γ ∈ (0; +∞). There exists a constant Cγ > 0 such that, for any n
X ∈ Mn satisfying X ∞ γ ,
sup P
x∈R

S
s

x − Φ(x)

Cγ

n log n

+ min
s3

1/3
,
1

V2−1

V2−1

1/2
∞

2 and any

.

Mourrat [11] generalized Corollary 1 and obtained the optimality of the result to any p ∈ [1; +∞).
Theorem 3. (See [11].) Let p ∈ [1; +∞) and γ ∈ (0; +∞). There exists a constant Cp,γ > 0 such that, for
any n 2 and any X ∈ Mn satisfying X ∞ γ ,
sup P
x∈R

S
s

x − Φ(x)

Cp,γ

n log n
+
s3

V2−1

p
p

+ s−2p

1/(2p+1)

.

The aim of this article is to extend these results to L1 -bounds in the mean central limit theorem for
martingale-difference sequences.
Theorem 4. Let 0 < α β < ∞, 0 < γ < ∞. If X 3 γ , σj2 = σ 2j a.s., and α
then there exists a constant C = C(α, β, γ) ∈ (0; ∞) such that
FS/s − Φ

Theorem 5. Let 0 < γ < ∞. If X
such that

∞

β for 1

j

n,

Cn−1/4 .

1

γ and V 2 (X) = 1 a.s., then there exists a constant 0 < C < ∞
FS/s − Φ

1

C

We have the following corollary, similar to Corollary 1.
Lith. Math. J., 54(1):48–60, 2014.

σ 2j

γ 3 n log n
.
s3


50

L.V. Dung, T.C. Son, and N.D. Tien

Corollary 2. Let 0 < γ < ∞ and p > 1/2. If X
depending only on p, such that

FS/s − Φ

γ 3 n log n
+ min
s3

C

1

γ , then there exists a positive constant C = C(p),

∞

V2−1

1/2
,
∞

E V2−1

p 1/2p

.

The following corollary is an L1 -version of Theorem 3.
Corollary 3. Let 0 < γ < ∞ and p > 1/2. If X
depending only on p, such that
FS/s − Φ

1

γ , then there exists a positive constant C = C(p),

∞

γ 3 n log n
p
+ E V 2 − 1 + s−2p
s3

C

1/2p

.

1/3

Note that the term V 2 − 1 1 appearing in Corollary 1 is replaced by the smaller term (E|V 2 − 1|p )1/2p
in Corollary 2, and the term ( V 2 − 1 pp + s−2p )1/(2p+1) appearing in Theorem 3 is replaced by the smaller
term (E|V 2 − 1|p + s−2p )1/2p in Corollary 3.

2

Auxiliary lemmas

For two random variables X and Y with distribution functions FX and GY , respectively, applying the
Kantorovich–Rubinstein theorem (see, e.g., [6, Thm. 11.8.2]), we have that

∞

FX − G Y

1

FX (x) − GY (x) dx = sup E f (X) − E f (Y ) ,

=

f ∈Λ1

−∞

where Λ1 is the set of 1-Lipschitzian functions from R to R. For more details, we refer the reader to [8] and [5].
For functions f, g : R → R, their convolution f ∗ g is defined by
∞

f ∗ g(x) =

f (x − y)g(y) dy.
−∞

We have the following lemmas.
p, q, r

Lemma 1. (See [3, p. 205].) If 1
have

∞, 1/p + 1/q = 1 + 1/r, f ∈ Lp (R), and g ∈ Lq (R), then we

f ∗g

f

r

p

g q.

Lemma 2. Let X and η be real random variables. Then, for any p > 1/2, we have
FX − Φ

1

FX+η − Φ

1

+ 2(2p + 1) E η 2p X

Proof. The conclusion is trivial in the case of E(η 2p | X)
γ < ∞. For any a > 0, we have that

∞

∞

FX − Φ

1

P(X

=
−∞

t − a) − Φ(t − a) dt

1/2p
.
∞

= ∞. So, we assume that E(η 2p | X)

(2.1)
∞

=


L1 bounds for some martingale CLT

51

∞

∞

P(X

t − a) − P(X + η

t) dt +

−∞

Φ(t) − Φ(t − a) dt

−∞

+ FX+η − Φ 1 .

(2.2)

First, we consider the first term on the right-hand side of (2.2). We have that
P(X + η

t) = E P(η

t − X | X)

E I(X

= P(X

t − a) − E I(X

t − a)P(η

t − X | X)

t − a)P(η > t − X | X) ,

where
E I(X

t − a)P(η > t − X | X)

γE (t − X)−2p I(X

t − a) .

γE (t − X)−2p I(X

t − a) ,

Therefore,
P(X

t − a) − P(X + η

t)

+

which implies
∞

P(X

t − a) − P(X + η

t)

+

dt

−∞
∞

∞

γE (t − X)

−2p

I(X

(t − X)−2p I(X

t − a) dt = γE

−∞

t − a) dt

−∞
∞

(t − X)−2p dt

= γE

= (2p − 1)

X+a

γ
a2p−1

.

(2.3)

On the other hand,
P(X + η

t) = E I(X

t − a)P(η

+ E I(t − a < X

t − X | X)
t + a)P(η

+ E I(X > t + a)P(η

t − X | X)

t − X | X)

P(X

t − a) + P(t − a < X

t + a) + E I(X > t + a)P(η

t − X | X)

P(X

t − a) + P(t − a < X

t + a) + γE (t − X)−2p I(X > t + a) ,

which implies that
P(X + η

t) − P(X

t − a)

P(t − a < X

Hence,
∞

P(X
−∞
Lith. Math. J., 54(1):48–60, 2014.

t − a) − P(X + η

t)

−

dt

t + a) + γE (t − X)−2p I(X > t + a) .


52

L.V. Dung, T.C. Son, and N.D. Tien
∞

∞

P(t − a < X

γE (t − X)−2p I(X > t + a) dt

t + a) dt +

−∞

−∞
∞

X−a

(t − X)

= 2a + γE

−2p

I(X > t + a) dt

(t − X)−2p dt

= 2a + γE

−∞

= 2a + (2p − 1)

−∞

γ
a2p−1

.

(2.4)

Combining (2.3) and (2.4) yields
∞

P(X

t − a) − P(X + η

t) dt

−∞
∞

∞

P(X

=

t − a) − P(X + η

−

t)

P(X

dt +

−∞

t − a) − P(X + η

t)

+

dt

−∞

2a + 2(2p − 1)

γ
a2p−1

.

(2.5)

Next, we consider the second term on the right-hand side of (2.2). We have that
∞

0

∞

Φ(t − a) − Φ(t) dt =
−∞

Φ(t − a) − Φ(t) dt +

−∞

Φ(t − a) − Φ(t) dt
0

0

∞

aϕ(t) dt +
−∞

aϕ(t − a) dt

2a.

(2.6)

0

Combining (2.2), (2.3), and (2.6) yields
FX − Φ

FX+η − Φ

1

1

+ 2 (2p − 1)

γ
a2p−1

+ 2a .

Taking a = γ 1/2p gives conclusion (2.1) of Lemma 2.
Lemma 3. Let ψ be a function R → R with ψ
E ψ(X)

∞

< ∞ and ψ

ψ

∞ |FX

∞

< ∞. If X is a random variable, then

− Φ|1 + ψ

∞.

Proof. It is clear that
∞

E ψ(X) − E ψ(N )

∞

ψ(x) dFX (x) −

=
−∞

∞

ψ(x) dΦ(x) =

−∞

FX (x) − Φ(x) ψ (x) dx
−∞

∞

ψ

∞
−∞

FX (x) − Φ(x) dx = ψ

∞

FX − Φ

1


L1 bounds for some martingale CLT

53

and
E ψ(N )

3

ψ

∞.

Proof of Theorem 4

Let Z1 , Z2 , . . . , Zn , η be independent normally distributed random variables with mean 0 and E(Zj2 ) = σ 2j ,
n
E(η 2 ) = n1/2 . Let Um = m−1
j=1 Xj /s and Z =
j=1 Zj . According to Lemma 2 with p = 1, we have
FS/s − Φ

1

F(S+η)/s − Φ

1

1/2

1
E η2
s2

+C

F(S+η)/s − F(Z+η)/s

1

F(S+η)/s − F(Z+η)/s

∞

1/4

+ Cn

1

+C

1
E η2
s

.

1/2
∞

(3.1)

On the other hand, by a proof is similar to that of Theorem 1 in [4] we get that
S+η
P
s

n

Z +η
t −P
s

t

|Xm |3
ϕ
λm s 3

E
m=1
n

E

+

m=1

|Zm |3
ϕ
λm s 3

where 0 θm , θm 1 and λm = ( nj=m+1 σ 2j + n1/2 )/s.
∞
Applying the Fubini theorem and noting that −∞ |ϕ (t)| dt
∞

S+η
s

P
−∞

∞

n

|Xm |3
ϕ
λ3m s3

E
m=1 −∞
n

∞

m=1 −∞
∞

n

E
m=1

−∞

|Xm |3
ϕ
λ3m s3
∞

n

E

+
m=1
n
m=1

−∞

|Xm |3
2E
λ3m s3

t

dt

+

t − Um
Zm
− θm
λm
λm s
2E

m=1

|Zm |3
λ3m s3

Combining (3.1) and (3.2) yields

The theorem is proved.
Lith. Math. J., 54(1):48–60, 2014.

1

dt

t − Um
Zm

− θm
λm
λm s

n

,

2, we have

t − Um
Xm
− θm
λm
λm s

|Zm |3
ϕ
λ3m s3

FS/s − Φ

t − Um
Zm
− θm
λm
λm s

t − Um
Xm

− θm
λm
λm s

|Zm |3
ϕ
λ3m s3

E

+

Z +η
s

t −P

t − Um
Xm
− θm
λm
λm s

Cn−1/4 .

dt

dt

dt

Cn−1/4 .

(3.2)


54

L.V. Dung, T.C. Son, and N.D. Tien

4

Proof of Theorem 5

For n ∈ N, s > 0, and γ > 0, let
G(s, γ) = X ∈ Mn : s(X) = s, X

γ, V 2 (X) = 1 a.s.

∞

and
Δ(n, s, γ) = sup sup E f
f ∈Λ1

S(X)
s

− E f (N) : X ∈ G(s, γ) .

It is clear that Δ(n, s, γ) Δ(n − 1, s, 2γ).
For a fixed element X ∈ G(s, γ), where we assume that γ 1, let Z1 , Z2 , . . . , Zn be i.i.d. standard normal
variables, and let η be a centered normal r.v. with variance κ2 such that η is independent of anything else. The
variance κ2 will be specified later, but in any case, κ2 > 2γ 2 .
Let
m−1
j=1 Xj

Um =

n
j=m+1 σj Zj

+η

λ2m

n
2
j=m+1 σj
s2

+ κ2

.
s
s
Conditioned on σ(Fn+1 , Zm ), Wm is normally distributed with mean 0 and variance λ2m , and Z =
n
j=1 σj Zj /s is a standard normal variable. Hence, by Lemma 2 we have

,

Wm =

FS/s − Φ

F(S+η)/s − F(

1

n
j=1

,

=

σj Zj +η)/s 1

κ
+C .
s

(4.1)

Now we consider the first term on the right-hand side of (4.1). Let ϕλm (x) be the density function of Wm .
For any 1-Lipschitzian f , according to an idea that goes back to Lindeberg [10], we write
E f

S+η

s

−E f

n
j=1 σj Zj

s

n

=

E f Wm + U m +

Xm
s

− E f Wm + U m +

E f ∗ ϕλ m U m +

Xm
s

− E f ∗ ϕλ m U m +

m=1
n

=
m=1
n

E gm Um +

=
m=1
n

E

=
m=1

+

+η

Xm
s

− E gm Um +

−

σm Z m
s

σm Z m

s

(where gm = f ∗ ϕλm )

2 Z2 − X2
σm
m
m
gm (Um )
s2

σm Z m − X m
gm (Um ) −
s

θm σm Z m
(σm Zm )3
gm Um −
3
s
s

σm Z m
s

3
Xm
θ Xm
gm Um − m
3

s
s

.

Since Um and λm are F m−1 -measurable, where F m−1 is the completion of Fm−1 , from E(Xm | F m−1 ) =
2 Z 2F
2
2
E(σm Zm | F m−1 ) = 0 a.s. and E(σm
j m−1 ) = E(Xj | F m−1 ) = σj a.s. it follows that the first two sums
in the above expression must vanish. Moreover, since gm (x) = f ∗ ϕλm (x), we get that
S+η
E f
s

−E f

n
j=1 σj Zj

s

+η


L1 bounds for some martingale CLT
n

|σm Zm |3

s3

E
m=1
n

1
6s3
+

:=

m=1

θσm Zm
s

E |σm Zm |3 f ∗ ϕλm Um −

m=1

+E

3|
|Xm
s3

gm Um +

θm Xm

s

θm Xm
s

E |Xm |3 f ∗ ϕλm Um −

n

1
6s3

gm Um +

55

θm σm Z m
s

1
(I + II).
6s3

(4.2)

We define the sequence of stopping times τj (1
k

τ0 = 0,

τj = inf k:

j

n) by

j
n

σi2

i=1

for 1

j

n − 1,

τn = n.

θm Xm
s

.

Then
n
m=1

E |Xm |3 f ∗ ϕλm Um −
τj

n

E

=

m=τj−1 +1

j=1

If τj−1 < m

θm Xm
s

|Xm |3 f ∗ ϕλm Um −

(4.3)

τj , then
λ2m

n
2
j=τj +1 σj
s2
n

λ2m

+ κ2

s2 − js2 /n − γ 2 + κ2
:= λ2j ,
s
s2 − (j − 1)s2 /n + κ2
2
:= λj .
2
s

σj2 + κ2 s2

j=τj−1 +1

We denote Rm =
be defined by

m−1
i=τj−1 +1 Xi

and Amt = {|Rm |
ψ(x) = sup

|Uτj−1 +1 − t|/2} for t ∈ R. Let the function ψ : R → R

ϕ (y) : |y|

|x|
−1 .
2

We conclude that, for every t,
ϕ

holds on Amt ∩ {τj−1 < m
τj

E
m=τj−1 +1
Lith. Math. J., 54(1):48–60, 2014.

U m − t θm Xm
−
λm
λm s

ψ

τm }. Then,
|Xm |3 f ∗ ϕλm Um −

θm Xm
s

Uτj−1 +1 − t
λj

56

L.V. Dung, T.C. Son, and N.D. Tien
τj

γE
m=τj−1 +1

2
Xm
f ∗ ϕλ m U m −
∞

τj

γE
m=τj−1 +1
τj

γE
m=τj−1 +1

2
Xm

2
Xm

∞

m=τj−1 +1

γE
m=τj−1 +1

∞

m=τj−1 +1

2
Xm

τj
m=τj−1 +1

2
Xm

ψ

Uτj−1 +1 − t
λj

−∞

f (t) dt

∞

m=τj−1 +1

m=τj−1 +1

U m − t θm Xm
−
f (t) IAcmt dt
λm
λm s

λ−3
m ϕ

−∞
∞

τj

τj

γλ−3
j E

U m − t θm Xm
−
f (t) IAmt dt
λm
λm s

λ−3
m ϕ

−∞

+ γE

+

−∞

2
Xm

τj

γλ−3
j E

U m − t θm Xm
−
f (t)IAcmt dt
λm
λm s

λ−3
m ϕ

∞

τj

+

2
Xm

θm Xm
− t f (t) dt
s

U m − t θm Xm
−
f (t)IAmt dt
λm
λm s

λ−3
m ϕ
−∞

+ γE

γλ−3
j E

ϕλ m U m −
−∞
∞

τj

γλ−3
j E

θm Xm
s

2
Xm

IAcmt dt

−∞
∞

2
Xm

ψ

Uτj−1 +1 − t
λj

−∞

f (t) dt

∞

τj
m=τj−1 +1

2
Xm

IAcmt dt

−∞

:= γλ−3
j (Mj + Nj ).

(4.4)

We first consider Mj . Since Uτj−1 +1 is Fτj−1 -measurable, we obtain
∞

Mj =

τj

E
m=τj−1 +1

−∞
∞

τj

E
−∞
∞

ψ

−∞

Uτj−1 +1 − t
λj

Uτj−1 +1 − t
λj

Uτj−1 +1 − t

m=τj−1 +1

E ψ

=

2
Xm
ψ

λj

f (t) dt

2
E Xm
Fτj−1

τj
m=τj−1 +1

2
E Xm
Fτj−1

f (t) dt

f (t) dt


L1 bounds for some martingale CLT
∞

2γ

2

Uτj−1 +1 − t

E ψ
∞

2

= 2γ E

f (t) dt

λj

−∞

Uτj−1 +1 − t

ψ

f (t) dt = 2γ 2 E gj (Uτj−1 +1 ) ,

λj

−∞

57

where
∞

gj (x) =

x−t
λj

ψ
−∞

f (t) dt.

Now, since
n

n

Xi2

E

Fτj−1

σi2 Fτj−1

=E

i=τj−1 +1

s2 1 −

i=τj−1 +1

j−1
n

a.s.,

from Lemma 3 we obtain

E gj (Uτj−1 +1 )

C

FS/s − Φ

1

j−1
+ λj .
n

+

1−

1−

j−1
+ λj .
n

Hence,
Mj

Cγ 2

FS/s − Φ

1

+

(4.5)

Next, we consider Nj . Let
k

Bj =

Xi >

max

τj−1
i=τj−1 +1

|Uτj−1 +1 − t|
.
2

Since Am is (Fm−1 ∨ Fτj−1 )-measurable, we have
∞

τj

Nj =

E

m=τj−1 +1

−∞

∞
2
σm
IAcm

∞

2γ

2

E min 1,

−∞
∞

2γ

2

E min 1,

−∞
∞

Cγ 2

E min 1,

−∞
Lith. Math. J., 54(1):48–60, 2014.

dt

2γ

2

P(Bj ) dt

−∞

|Uτj−1 +1 − t|

−2

2λj
|Uτj−1 +1 − t|
2λj
|Uτj−1 +1 − t|
2λj

−2

2

k

E

Xi

max

τj−1 τj

E

i=τj−1 +1

Xi2

Fτj−1

i=τj−1 +1
−2

Fτj−1

dt = Cγ 2 E hj (Uτj−1 +1 ) ,

dt

dt

58

L.V. Dung, T.C. Son, and N.D. Tien

where
∞

hj (x) =

−2

x−t
2λj

min 1,
−∞

dt.

By Lemma 2 we obtain
Cγ 2

Nj

FS/s − Φ

1−

+

1

j−1
+ λj .
n

(4.6)

Combining (4.3), (4.4), and (4.5) with (4.6) yields
C γ 3 n κ2 − 2γ 2

I

−1/2

Δ(n, s, γ) + γ 3 n log n .

(4.7)

Next, we need to derive a bound for II on the right-hand side of (4.2).
For t ∈ R, put
A˜mt =

|Uτj−1 +1 − t|
,
4

|Rm |

Bmt =

|Uτj−1 +1 − t|
.
8

σm |Zm |

Then
τj

E
m=τj−1 +1

3
σm
|Zm |3 f ∗ ϕλm Um −

θm σn Z m
s

∞

τj

E
m=τj−1 +1

3
σm

|Zm |3

τj

E
m=τj−1 +1

ϕλ m U m −
−∞

3 |Z |3
σm
m
3
λm

τj

+E
m=τj−1 +1

∞

ϕ
−∞

3 |Z |3
σm
m
λ3m

θm σn Z m
− t f (t) dt
s

Um − t θm σn Zm /s
−
f (t)IAmt ∩Bmt dt
λm
λm

∞

τj

IA˜cmt dt

+E
m=τj−1 +1

−∞

3 |Z |3
σm
m
λ3m

∞
c
IBmt

dt .

−∞

Making use of the independence of random variables {Zm }, the first and second sums can be estimated as
above. As for the third sum, note that
τj

E
m=τj−1 +1

3 |Z |3
σm
m
3
λm s 3

∞
c
IBmt
dt

γλ−3
j E

−∞

τj
m=τj−1 +1

∞
2
σm
|Zm |3
−∞

cγ 3 λ−3
j E ψ (Uτj−1 +1 ) ,

where
∞

ψ (x) =

g
−∞

x−t
λj

dt

I{8|Zm |>λ−1 |Uτ
j

j−1 +1

−t|}

dt

L1 bounds for some martingale CLT

59

and
∞
3

g (x) = E |Zm | I{|Zm |

t2 P |Zm | > t dt + |x|3 P |Zm | > |x| .

=2

|x|}

|x|

Using this expression as above, we obtain
C γ 3 n κ2 − 2γ 2

II

−1/2

Δ(n, s, γ) + γ 3 n log n .

Combining this with (4.1), (4.2), and (4.7), we obtain

Cs−3 γ 3 n κ2 − 2γ 2

Δ(n, s, γ)

−1/2

Taking κ2 = 2γ 2 + C 2 22 γ 6 n2 and putting Kn = supγ

κ
Δ(n − 1, s, 2γ) + γ 3 n log n + C .
s

1, 0
1
Kn−1 + C
2

for n

lim sup Kn

C.

Kn

(4.8)

Δ(n, s, γ)/γ 3 s−3 n log n, from (4.8) we get
4

and, hence,
n→∞

Therefore, the theorem is proved.

5

Proof of corollaries

Let X ∈ Mn with X ∞ γ , and let a = s2 V 2 (X) − s2 ∞ . We define Xn+1 , . . . , Xn+[2a/γ 2 ]+1 as in the
proof of Corollary 1 in [4].
By using the triangle inequality and Lemma 3 we have the following inequalities:
FS/s − Φ

1

sˆ
F −Φ
s S/ˆs
sˆ
Fˆ − Φ
s S/ˆs

+ FsˆN/s − Φ 1
√
a
sˆ
(by Lemma 3)
+C

−1
1+C
s
s
√
sˆ n
ˆ
a
sˆ
ˆ γ 3 log n
C
+
(by Theorem 2)
+
−1
3
s
sˆ
s
s
√
√
γ 3 n log n
a
a
if
C
+
1 and n is sufficiently large.
3

s
s
s
1

Therefore, in this case, we obtain
FS/s − Φ

1

√
γ 3 n log n
a
C
+
.
3
s
s

(5.1)

√
Estimate (5.1) is also true for all n and a/s > 1 if C is suitably chosen.
ˆ =
For the estimate V 2 − 1 1 , we again let X = (X1 , . . . , Xn ) ∈ Mn with X
γ and define X
ˆ
ˆ
(X1 , . . . , X2n ) as in the proof of Corollary 1 in [4].

Then, by Theorem 2, FS/s
− Φ
Cγ 3 n log n/s3 , and by Burkholder’s inequality (see, e.g.,
ˆ
Lith. Math. J., 54(1):48–60, 2014.


60

L.V. Dung, T.C. Son, and N.D. Tien

[9, Thm. 2.11]) it is easy to see that
E(Sˆ − S)2p

p

CE s2 V 2 − s2 .

(5.2)

For any x > 0, we have
∞

FS/s − Φ

1

FS/s
ˆ −Φ

1

+
−∞

Sˆ
P
s

C
E(Sˆ − S)2p + 2x
x2p−1 s2p
E|V 2 − 1|p
FS/s
−
Φ
+
C
+ 2x,
1
ˆ
x2p−1
FS/s
ˆ −Φ

1

∞

S

t+x −P
s

+

t

dt +

Φ(t + x) − Φ(t) dt

−∞

(by (2.5) and (2.6))

(5.3)

and now putting x = (E V 2 − 1 )1/2p , we have
|FS/s − Φ|1

C

γ 3 n log n
+C E V2−1
s3

Combining this with (5.1) yields the conclusion of Corollary 2.
Substituting, instead of (5.2), the inequality E(Sˆ − S)2p
immediately get the conclusion of Corollary 3.

1/2p

.

C(E|s2 V 2 − s2 |p + 2γ 2p ) into (5.3), we

Acknowledgment. We would like to express our gratitude to the referee for his/her detailed comments and
valuable suggestions, which helped us to improve the manuscript.

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