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Rate of convergence in a theorem of Heyde
Statistics and Probability Letters 82 (2012) 1576–1582
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Rate of convergence in a theorem of Heyde Tingfan Xie ∗ , Jianjun He Department of Mathematics, China Jiliang University, Hangzhou 310018, China
article
abstract
info
Article history: Received 6 September 2011 Received in revised form 28 December 2011 Accepted 26 March 2012 Available online 21 April 2012 MSC: 60F15 60G50
Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with mean zero, and set n 2 S n = k=1 Xk , TX (t ) = EX I (|X | > t ). Heyde (1975) proved precise asymptotics for ∞ n=1 P (|Sn | ≥ nϵ) as ϵ ↘ 0. In this paper, we obtain a convergence rate in a theorem of Heyde (1975) under a second moment assumption only. Furthermore, under the additional assumption of TX (t ) = O(t −δ ) as t → ∞ for some δ > 0, we obtain a refined result. © 2012 Elsevier B.V. All rights reserved.
Keywords: Theorem of Heyde Rate of approximation I.i.d random variable Precise asymptotics
1. Introduction Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables. It is well-known, Hsu and Robbins (1947) introduced the concept of complete convergence, and proved that ∞
P (|Sn | > nϵ) < ∞,
for all ϵ > 0,
n =1
if EX = 0,
EX 2 = σ 2 < ∞.
(1.1)
A bit later, Erdös (1949, 1950) proved the converse. Denote λX (ϵ) = n=1 P (|Sn | ≥ nϵ), Heyde (1975) studied the behavior of λX (ϵ) as ϵ → 0, and showed the following Theorem A which is now called precise asymptotics, and is quoted extensively in Gut and Spˇataru (2000), Liu and Lin (2006), Jiang and Zhang (2007).
∞
Theorem A. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with mean zero and EX 2 = σ 2 < ∞. Then lim ϵ 2 λX (ϵ) = σ 2 .
ϵ→0
(1.2)
Theorem A means σ 2 can be approximated by ϵ 2 λX (ϵ) as ϵ → 0. Klesov (1994) studied the rate of the approximation of σ 2 by ϵ 2 λX (ϵ) under the condition E |X |3 < ∞, and proved the following result.
∗
Corresponding author. E-mail addresses: [email protected] (T. Xie), [email protected] (J. He).
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.03.034
T. Xie, J. He / Statistics and Probability Letters 82 (2012) 1576–1582
1577
Theorem B. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with mean zero, EX 2 = σ 2 > 0 and E |X |3 < ∞. Then lim ϵ
3/2
ϵ→0
σ2 λX (ϵ) − 2 ϵ
= 0.
(1.3)
Obviously, (1.3) can be expressed by ϵ 2 λX (ϵ) − σ 2 = o(ϵ 1/2 ), as ϵ → 0. Recently, He and Xie (in press) have improved Theorem B by the following Theorem C. Theorem C. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with mean zero, and 0 < δ ≤ 1. If EX 2 = σ 2 > 0
and
E |X |2+δ < ∞,
then
ϵ λX (ϵ) − σ = 2
2
O(ϵ), o(ϵ δ ),
δ = 1, 0 < δ < 1.
Gut and Steinebach (2012) extended the results of Klesov (1994) to a large class of convergence rate statements in precise asymptotics of partial sums. The aim of the present paper is to discuss the rate of approximation of σ 2 by ϵ 2 λX (ϵ) under the second moment only. Denote TX (t ) = EX 2 I (|X | ≥ t ). Obviously, EX 2 < ∞ if and only if limt →∞ TX (t ) = 0. We shall describe the rate of convergence in the theorem of Heyde by TX (t ). Especially, if the condition TX (t ) = O(t δ ) is satisfied for some positive constant δ > 0, then we can obtain a more refined result. Throughout this paper, C stands for a positive constant whose value may differ from one place to another. The paper is organized as follows. We shall give main results in Section 2, and provide the proofs in Section 3. 2. Main results Theorem 2.1. Let {X , Xn ; n ≥ 1} be a sequence of i.i.d. random variables with mean zero and EX 2 = σ 2 > 0. Then
ϵ 2 λX (ϵ) − σ 2 = O σ 3/2 ϵ 1/2 + TX (σ 3/2 ϵ −1/2 ) log
σ2 3 / 2 − 1 / TX (σ ϵ 2 ) + σ 3/2 ϵ 1/2
,
as ϵ → 0.
Remark 2.1. If EX 2 = σ 2 < ∞, we have limv→∞ TX (v) = 0. Then limϵ→0 TX (σ 3/2 ϵ −1/2 ) log result of Theorem A is covered by Theorem 2.1.
σ2
TX (σ 3/2 ϵ −1/2 )+σ 3/2 ϵ 1/2
= 0. The
Remark 2.2. The rate of approximation of σ 2 by ϵ 2 λX (ϵ) is not faster than O(ϵ 1/2 ) from Theorem 2.1; however, if {X , Xn ; n ≥ 1} be a sequence of i.i.d Gaussian random variables, He and Xie (in press) and Gut and Steinebach (2012) proved that the rate of approximation of σ 2 by ϵ 2 λX (ϵ) is faster than O(ϵ 1/2 ). Hence, in order to get a much faster rate of convergence, it is meaningful to discuss the rate of approximation under the additional assumption of TX (v). Theorem 2.2. Let {X , Xn ; n ≥ 1} be a sequence of i.i.d. random variables with mean zero. If TX (t ) = O(t −δ ) as t → ∞ for some δ > 0, and EX 2 = σ 2 > 0, then
2−δ δ ϵ ), O(σ σ 2 2 ϵ λX (ϵ) − σ = O σ ϵ log +1 , ϵ O(σ ϵ),
0 < δ < 1,
δ = 1, δ > 1.
Remark 2.3. The condition E |X |2+δ < ∞ for some δ > 0 implies TX (t ) ≤ t −δ E |X |2+δ I (|X | > t ) = o(t −δ ) = O(t −δ ). Comparing Theorem C with Theorem 2.2, when 0 < δ ≤ 1, the convergence rate in Theorem C is faster than that in Theorem 2.2. However, the results of Theorem 2.2 are obtained under more weaker conditions. On the other hand, if TX (t ) = O(t −α ) for 0 < δ < α < 1, by Theorem 2.2, ϵ 2 λX (ϵ) − σ 2 = O(ϵ α ) = o(ϵ δ ). Remark 2.4. If TX (t ) is regularly varying with index −δ at infinity for some δ > 0, then TX (t ) = t −δ l(t ), l(t ) is a slowly varying function at infinity. By the process similar to the proof of Theorem 2.2, and replacing t −δ by t −δ l(t ), we can obtain
ϵ 2 λ(ϵ) − σ 2 =
1 δ , O ϵ l ϵ 2 O ϵ+ϵ O(ϵ),
1/ϵ
1
And the details of the proof are omitted.
0 < δ < 1, l(t ) t
dt
,
δ = 1, δ > 1.
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T. Xie, J. He / Statistics and Probability Letters 82 (2012) 1576–1582
3. Proofs of main results First, we give some lemmas, which will be used in the following proofs. Lemma 3.1 is from He and Xie (in press). Lemma 3.3 is from Fuk and Nagaev (1971). Φ (x) is the standard normal distribution function, Φ (x) = √1
2π
φ(x) = Φ ′ (x).
x −∞
e−
t2 2
dt,
Lemma 3.1. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. standard normal distribution random variables. Then
ϵ 2 λX (ϵ) = ϵ 2
∞
2
√
2π
n =1
ϵ2 2 e−t /2 dt = 1 − + O(ϵ 3 ), 2 ϵ n ∞
as ϵ → 0.
√
If {Xn , n ≥ 1} is a sequence of independent random variables with mean zero and finite variance. Put EXj2 = σj2 , Bn =
σj2 . Bikelis (1966) proved the following inequality: √ n n (1+|x|) Bn 1 −3/2 −3 Xj < x − Φ (x) ≤ CBn (1 + |x|) u2 dVj (u)dv P √ Bn j=1 |u|>v j =1 0
n
j =1
for every x, where Vj (x) = P (Xj < x) is the distribution function of random variable Xj . By applying this inequality to a √ sequence of i.i.d. random variables with mean zero and variance 1, let x = ϵ n, we get the following. Lemma 3.2. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with mean zero and EX 2 = 1. Then for any ϵ > 0,
∞ (1+ϵ √n)√n C −t 2 / 2 P (|Sn | > nϵ) − √2 e dt ≤ TX (v)dv. √ √ √ ( 1 + ϵ n) 3 n 0 2π ϵ n Lemma 3.3. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with E |X |β < ∞, where 1 < β ≤ 2. For x > 0, y > 0, we have P (|Sn | > x) ≤ nP (|X | ≥ y) + 2nx/y
eE |X |β
x/y
xyβ−1
.
Proof of Theorem 2.1. We prove Theorem 2.1 in two cases: σ 2 = 1, and σ 2 ̸= 1. (a) σ 2 = 1, assume that 0 < ϵ < 1, note that ∞ 2ϵ 2 ∞ −t 2 /2 ϵ 2 λX (ϵ) = I + √ e dt , √ 2π n=1 ϵ n
where I = ϵ2
∞ n=1
2 P (|Sn | > nϵ) − √ 2π
∞
2 e−t /2 dt √
ϵ n
.
Applying Lemma 3.1, we get
ϵ 2 λX (ϵ) = I + 1 −
ϵ2 2
+ O(ϵ 3 ),
as ϵ → 0.
(3.1)
By (3.1), in order to prove Theorem 2.1, we only need to show that
I = O ϵ 1/2 + TX (ϵ −1/2 ) log
1 TX (ϵ −1/2 ) + ϵ 1/2
,
as ϵ → 0.
We write
I = ϵ 2
K
ϵ2
n=1
+ϵ
2
∞ 2 P (|Sn | > nϵ) − √
n= K2 +1 ϵ
2π
∞ √ ϵ n
e
−t 2 /2
dt
=: I1 + I2 ,
T. Xie, J. He / Statistics and Probability Letters 82 (2012) 1576–1582
1579
where [x] denotes the integer part of x, and K is a large enough positive integer which will be determined later. First, we shall deal with the term I1 , by Lemma 3.2,
| I1 | ≤ ϵ
2
K
ϵ2
√ √ (1 + ϵ n)3 n
n=1
3/4
1
ϵ2
2 = ϵ
+ϵ
√ √ (1+ϵ n) n
C
1
ϵ2
2
n=1
n=
1 ϵ2
TX (t )dt
0
+ϵ
3/4
2 n=
+1
K
(1+ϵ ϵ2 C √ √ (1 + ϵ n)3 n 0 1 ϵ2
√ √ n) n
TX (t )dt
+1
= : I11 + I12 + I13 . Since TX (t ) ≤ EX 2 = 1, and TX (t ) is a nonnegative decreasing function, then
3/4
1
ϵ2
C ≤ C ϵ 1/2 = O(ϵ 1/2 ).
I11 ≤ ϵ 2
(3.2)
n =1
For I12 , noting that ( ϵ12 )3/4 < n < ϵ12 implies √1ϵ < nϵ < 1ϵ , we can obtain √1ϵ < (1 + ϵ
I12 ≤ ϵ
2 n=
1 ϵ2
≤ϵ
1
ϵ2
n=
C
1
√ √ +ϵ n ϵ
dt
ϵ2
2
+1
1
n=
1 ϵ2
√1 ϵ
+1
3/4
1
√ ϵ
0
1 ϵ2
1 ϵ2
dt + TX
n
2
√
3/4
√ √ (1+ϵ n) n
√
1/ ϵ
C
√ √ n) n ≤ 2ϵ . Thus,
3/4
C
√ TX ϵ n
1
√
ϵ
+1
1 = O ϵ 2 + TX (ϵ −1/2 ) .
(3.3)
Analogously,
I13 ≤ ϵ
K
ϵ2
n=
1 ϵ2
K
3
n=1+
≤ C ϵ
1
1
ϵ 3 n2
+ C ϵ 2 TX
1
√
ϵ
K
1 ϵ2
1
ϵ 3 n2
+ TX
1
√ ϵ
dt
√
1/ ϵ
1
ϵ2
√ (1 + ϵ n)2
1
n=1+
√ √ (1+ϵ n) n
1
K
√ ϵ
ϵ2
ϵ2
n=1+
n=1+
1
K
ϵ2
ϵ2
3 2
0
ϵ2
≤ Cϵ 2
dt + TX
√ √ ( 1 + ϵ n) 3 n
+1
√
1/ ϵ
C
2
1 ϵ2
n
1 = O ϵ 2 + TX (ϵ −1/2 ) log K .
(3.4)
From (3.2) to (3.4), we get I1 = O ϵ 1/2 + TX (ϵ −1/2 ) log K .
(3.5)
Next, we estimate the term I2 , using the following inequalities 1 − Φ (x) ≤ 2
√
2π
√
φ(x)
√ 2φ(ϵ n) 2 e−t /2 dt = 2(1 − Φ (ϵ n)) ≤ = O(ϵ −5 n−5/2 ). √ √ ϵ n ϵ n
∞
x
, x ∈ R+ and t 2 e−t ≤ 1, t > 2. We can obtain
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T. Xie, J. He / Statistics and Probability Letters 82 (2012) 1576–1582
So, I2 = ϵ
∞
2
P (|Sn | > nϵ) − ϵ
∞
∞ n=
K ϵ2
2π
n= K2 +1 ϵ
= ϵ2
√
∞
P (|Sn | > nϵ) + C ϵ 2
∞
2
n= K2 +1 ϵ
n= K2 +1 ϵ
≤ ϵ2
∞
2
√
2 e−t /2 dt
ϵ n
ϵ −5 n−5/2
n= K2 +1 ϵ
P (|Sn | > nϵ) + O(K −3/2 ).
(3.6)
+1
Using the method of Heyde (1975) to estimate the first term of (3.6), and specializing Lemma 3.3 by x = 2y = nϵ and β = 2, we have P (|Sn | > nϵ) ≤ nP
|X | ≥
1 2
nϵ
+ 8e2 n−2 ϵ −4 .
Hence ∞
I2 ≤ ϵ 2
nP
1
|X | >
2
n= K2 +1 ϵ
∞
= ϵ2 n=
K ϵ2
n +1
∞
= ϵ2
|x|≥ 12 nϵ
n
n= K2 +1 ϵ
∞
= ϵ2
k= K2 +1 ϵ
∞
≤C k=
K ϵ2
dFX (x) + O(K −1 )
1 kϵ≤|x|< 12 (k+1)ϵ 2
dFX (x) + O(K −1 )
dFX (x)
k
1 kϵ≤|x|< 12 (k+1)ϵ 2
dFX (x) + O(K −1 )
≤C |x|≥ 2Kϵ
n + O(K −1 )
n=1
+1
ϵ −4 n−2 + O(K −3/2 )
n= K2 +1 ϵ
1 kϵ≤|x|< 12 (k+1)ϵ 2
(kϵ)2
∞
+ 8e2 ϵ 2
∞ k=n
nϵ
x2 dFX (x) + O(K −1 ) = O TX
K
2ϵ
+ K −1 .
(3.7)
Together (3.5) with (3.7), we get
I = O ϵ 1/2 + K −1 + TX (ϵ −1/2 ) log K + TX
K
2ϵ
.
Set K = [ T (ϵ −1/12 )+ϵ 1/2 ], then X K lim √ = +∞,
ϵ→0
ϵ
TX (t ) is a nonnegative decreasing function, then for sufficiently small ϵ > 0, we have
TX
K 2ϵ
= TX
K
√ ϵ −1/2 2 ϵ
≤ TX (ϵ −1/2 ).
Therefore
I = O ϵ 1/2 + TX (ϵ −1/2 ) log
1 TX (ϵ −1/2 ) + ϵ 1/2
,
as ϵ → 0.
Then
ϵ 2 λX (ϵ) = 1 + O ϵ 1/2 + TX (ϵ −1/2 ) log
1 TX (ϵ −1/2 ) + ϵ 1/2
,
as ϵ → 0.
T. Xie, J. He / Statistics and Probability Letters 82 (2012) 1576–1582
1581
(b) σ 2 ̸= 1, then
∞ ϵ 2 ϵ |Sn | ϵ λX (ϵ) − σ = σ P −1 ≥n σ n=1 σ σ = σ 2 O σ −1/2 ϵ 1/2 + TX /σ (σ 1/2 ϵ −1/2 ) log 2
2
2
= O σ 3/2 ϵ −1/2 + TX (σ 3/2 ϵ −1/2 ) log Thus, the proof of Theorem 2.1 is now complete.
1 TX /σ (σ 1/2 ϵ −1/2 ) + σ −1/2 ϵ 1/2
σ2 3 / 2 − 1 / TX (σ ϵ 2 ) + σ 3/2 ϵ 1/2
.
Proof of Theorem 2.2. Without the loss of generality, we assume that σ 2 = 1, 0 < ϵ < proof of Theorem 2.1, we have
ϵ2 ϵ 2 λX (ϵ) − 1 = I − + O(ϵ 3 ) 2 1 3 ∞ ϵ 2 2 2 = ϵ +ϵ P (|Sn | > nϵ) − √ n=1
n=
1
ϵ3
2π
+1
∞
√ ϵ n
2
e
− t2
dt
1 . 2
By the process similar to the
+ O(ϵ 2 )
= : I1 + I2 + O(ϵ 2 ). First, we estimate the term I2 . It is easily seen that ∞
I2 ≤ ϵ 2 n=
1
ϵ3
nP
n=
1
ϵ3
∞
+ 8e2 ϵ 2 n=
|x|≥ 12 nϵ
+1
1 2ϵ 2
+ϵ
1
ϵ3
∞
ϵ − 4 n− 2 + C ϵ 2 +1
n=
1
ϵ3
ϵ −5 n−3/2 +1
dFX (x) + O(ϵ) + O(ϵ 3/2 )
n
= O TX
2
nϵ
+1
∞
= ϵ2
1
|X | >
1
= O TX
ϵ
+ϵ
= O(ϵ δ + ϵ).
(3.8)
For I1 , we have
1
ϵ3
C
√ √ (1+ϵ n) n
TX (t )dt √ √ (1 + ϵ n)3 n 0 1 1 (1+ϵ √n)√n ϵ2 ϵ3 C 2 2 = ϵ +ϵ TX (t )dt √ √ (1 + ϵ n)3 n 0 n =1 1
| I1 | ≤ ϵ
2
n=1
n=
+1
ϵ2
= : I11 + I12 .
(3.9)
For I11 , since TX (t ) ≤ EX 2 = 1, TX (t ) = O(t −δ ), and noting that n ≤ ϵ12
I11 = ϵ 2
1
ϵ2
n =1
√ √ (1+ϵ n) n
C
√
( 1 + ϵ n)
√ 3
n
√ √ √ implies (1 + ϵ n) n ≤ 2 n ≤ 2ϵ , we can get
TX (t )dt
0
1 ϵ2 C 2 ≤ϵ √ 1+
n =1
n
≤ Cϵ 1 +
2/ϵ
TX (t ) dt
1 2/ϵ
t
−δ
dt
1
1 = O(ϵ δ + ϵ)I (δ ̸= 1) + O ϵ log + 1 I (δ = 1) . ϵ
(3.10)
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T. Xie, J. He / Statistics and Probability Letters 82 (2012) 1576–1582
For I12 , noting that ϵ12 ≤ n ≤ ϵ13 implies ϵ
I12 = ϵ
ϵ3
2 n=
1 ϵ2
1
+1
n=
√ √ (1 + ϵ n)3 n
1
+1
√ (1 + ϵ n)3 n
ϵ3
2 n=
1 ϵ2
2nϵ
√
1 ϵ2
C +1
n2 ϵ 3
1+
TX (t )dt
0
2nϵ
TX (t )dt
0
C
n > 1 and nϵ ≤ ϵ12 , it follows
√ √ (1+ϵ n) n
C
ϵ3
≤ ϵ2
≤ϵ
1
√
t
−δ
dt
1
1
δ
= O(ϵ + ϵ)I (δ ̸= 1) + O ϵ log
ϵ
+ 1 I (δ = 1) .
(3.11)
By (3.8)–(3.11), we obtain that
O(ϵ δ ), 1 2 ϵ λX (ϵ) − 1 = O ϵ log +1 , ϵ O(ϵ),
0 < δ < 1,
δ=1 δ > 1.
If σ 2 ̸= 1, note that λX (ϵ) = λ X ( σϵ ), we have σ
ϵ 2 λX (ϵ) − σ 2 = σ 2
ϵ 2 σ
λX σ
ϵ σ
2−α α ϵ ), O(σ σ 2 − σ = O σ ϵ log +1 , ϵ O(σ ϵ),
The proof of Theorem 2.2 is now complete.
0 < δ < 1,
δ = 1, δ > 1.
Acknowledgments The authors are grateful to anonymous referees and the editor for their constructive comments and suggestions, which have improved the presentation of the paper. References Bikelis, A., 1966. Estimates of the remainder in the central limit theorem. Litovsk. Math. Sb. 6, 323–346. Erdös, P., 1949. On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 286–291. Erdös, P., 1950. Remark on my paper on a theorem of Hsu and Robbins. Ann. Math. Statist. 21, 138. Fuk, D.H., Nagaev, S.V., 1971. Probability inequalities for sums of independent random variables. Theory Probab. Appl. 16, 643–660. Gut, A., Spˇataru, A., 2000. Precise asymptotics in the Baum–Katz and Davis law of large numbers. J. Math. Anal. Appl. 248, 233–246. Gut, A., Steinebach, J., 2012. Convergence rates in precise asymptotics. J. Math. Anal. Appl. 390, 1–14. Hsu, P.L., Robbins, H., 1947. Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. USA 33, 25–31. Heyde, C.C., 1975. A supplement to the strong law of large numbers. J. Appl. Probab. 12, 903–907. He, J.J., Xie, T.F., 2012. Asymptotic property for some series of probability, Acta Math. Appl. Sin. http://dx.doi.org/10.1007/s10255-012-0138-6 (in press). Jiang, Y., Zhang, L.X., 2007. Precise rates in the law of logarithm for the moment convergence of i.i.d. random variables. J. Math. Anal. Appl. 327, 695–714. Klesov, O.I., 1994. On the convergence rate in a theorem of Heyde. Theory Probab. Math. Statist. 49, 83–87. Liu, W.D., Lin, Z.Y., 2006. Precise asymptotics for a new kind of complete moment convergence. Statist. Probab. Lett. 76, 1787–1799.
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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Rate of convergence in a theorem of Heyde Tingfan Xie ∗ , Jianjun He Department of Mathematics, China Jiliang University, Hangzhou 310018, China
article
abstract
info
Article history: Received 6 September 2011 Received in revised form 28 December 2011 Accepted 26 March 2012 Available online 21 April 2012 MSC: 60F15 60G50
Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with mean zero, and set n 2 S n = k=1 Xk , TX (t ) = EX I (|X | > t ). Heyde (1975) proved precise asymptotics for ∞ n=1 P (|Sn | ≥ nϵ) as ϵ ↘ 0. In this paper, we obtain a convergence rate in a theorem of Heyde (1975) under a second moment assumption only. Furthermore, under the additional assumption of TX (t ) = O(t −δ ) as t → ∞ for some δ > 0, we obtain a refined result. © 2012 Elsevier B.V. All rights reserved.
Keywords: Theorem of Heyde Rate of approximation I.i.d random variable Precise asymptotics
1. Introduction Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables. It is well-known, Hsu and Robbins (1947) introduced the concept of complete convergence, and proved that ∞
P (|Sn | > nϵ) < ∞,
for all ϵ > 0,
n =1
if EX = 0,
EX 2 = σ 2 < ∞.
(1.1)
A bit later, Erdös (1949, 1950) proved the converse. Denote λX (ϵ) = n=1 P (|Sn | ≥ nϵ), Heyde (1975) studied the behavior of λX (ϵ) as ϵ → 0, and showed the following Theorem A which is now called precise asymptotics, and is quoted extensively in Gut and Spˇataru (2000), Liu and Lin (2006), Jiang and Zhang (2007).
∞
Theorem A. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with mean zero and EX 2 = σ 2 < ∞. Then lim ϵ 2 λX (ϵ) = σ 2 .
ϵ→0
(1.2)
Theorem A means σ 2 can be approximated by ϵ 2 λX (ϵ) as ϵ → 0. Klesov (1994) studied the rate of the approximation of σ 2 by ϵ 2 λX (ϵ) under the condition E |X |3 < ∞, and proved the following result.
∗
Corresponding author. E-mail addresses: [email protected] (T. Xie), [email protected] (J. He).
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.03.034
T. Xie, J. He / Statistics and Probability Letters 82 (2012) 1576–1582
1577
Theorem B. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with mean zero, EX 2 = σ 2 > 0 and E |X |3 < ∞. Then lim ϵ
3/2
ϵ→0
σ2 λX (ϵ) − 2 ϵ
= 0.
(1.3)
Obviously, (1.3) can be expressed by ϵ 2 λX (ϵ) − σ 2 = o(ϵ 1/2 ), as ϵ → 0. Recently, He and Xie (in press) have improved Theorem B by the following Theorem C. Theorem C. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with mean zero, and 0 < δ ≤ 1. If EX 2 = σ 2 > 0
and
E |X |2+δ < ∞,
then
ϵ λX (ϵ) − σ = 2
2
O(ϵ), o(ϵ δ ),
δ = 1, 0 < δ < 1.
Gut and Steinebach (2012) extended the results of Klesov (1994) to a large class of convergence rate statements in precise asymptotics of partial sums. The aim of the present paper is to discuss the rate of approximation of σ 2 by ϵ 2 λX (ϵ) under the second moment only. Denote TX (t ) = EX 2 I (|X | ≥ t ). Obviously, EX 2 < ∞ if and only if limt →∞ TX (t ) = 0. We shall describe the rate of convergence in the theorem of Heyde by TX (t ). Especially, if the condition TX (t ) = O(t δ ) is satisfied for some positive constant δ > 0, then we can obtain a more refined result. Throughout this paper, C stands for a positive constant whose value may differ from one place to another. The paper is organized as follows. We shall give main results in Section 2, and provide the proofs in Section 3. 2. Main results Theorem 2.1. Let {X , Xn ; n ≥ 1} be a sequence of i.i.d. random variables with mean zero and EX 2 = σ 2 > 0. Then
ϵ 2 λX (ϵ) − σ 2 = O σ 3/2 ϵ 1/2 + TX (σ 3/2 ϵ −1/2 ) log
σ2 3 / 2 − 1 / TX (σ ϵ 2 ) + σ 3/2 ϵ 1/2
,
as ϵ → 0.
Remark 2.1. If EX 2 = σ 2 < ∞, we have limv→∞ TX (v) = 0. Then limϵ→0 TX (σ 3/2 ϵ −1/2 ) log result of Theorem A is covered by Theorem 2.1.
σ2
TX (σ 3/2 ϵ −1/2 )+σ 3/2 ϵ 1/2
= 0. The
Remark 2.2. The rate of approximation of σ 2 by ϵ 2 λX (ϵ) is not faster than O(ϵ 1/2 ) from Theorem 2.1; however, if {X , Xn ; n ≥ 1} be a sequence of i.i.d Gaussian random variables, He and Xie (in press) and Gut and Steinebach (2012) proved that the rate of approximation of σ 2 by ϵ 2 λX (ϵ) is faster than O(ϵ 1/2 ). Hence, in order to get a much faster rate of convergence, it is meaningful to discuss the rate of approximation under the additional assumption of TX (v). Theorem 2.2. Let {X , Xn ; n ≥ 1} be a sequence of i.i.d. random variables with mean zero. If TX (t ) = O(t −δ ) as t → ∞ for some δ > 0, and EX 2 = σ 2 > 0, then
2−δ δ ϵ ), O(σ σ 2 2 ϵ λX (ϵ) − σ = O σ ϵ log +1 , ϵ O(σ ϵ),
0 < δ < 1,
δ = 1, δ > 1.
Remark 2.3. The condition E |X |2+δ < ∞ for some δ > 0 implies TX (t ) ≤ t −δ E |X |2+δ I (|X | > t ) = o(t −δ ) = O(t −δ ). Comparing Theorem C with Theorem 2.2, when 0 < δ ≤ 1, the convergence rate in Theorem C is faster than that in Theorem 2.2. However, the results of Theorem 2.2 are obtained under more weaker conditions. On the other hand, if TX (t ) = O(t −α ) for 0 < δ < α < 1, by Theorem 2.2, ϵ 2 λX (ϵ) − σ 2 = O(ϵ α ) = o(ϵ δ ). Remark 2.4. If TX (t ) is regularly varying with index −δ at infinity for some δ > 0, then TX (t ) = t −δ l(t ), l(t ) is a slowly varying function at infinity. By the process similar to the proof of Theorem 2.2, and replacing t −δ by t −δ l(t ), we can obtain
ϵ 2 λ(ϵ) − σ 2 =
1 δ , O ϵ l ϵ 2 O ϵ+ϵ O(ϵ),
1/ϵ
1
And the details of the proof are omitted.
0 < δ < 1, l(t ) t
dt
,
δ = 1, δ > 1.
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3. Proofs of main results First, we give some lemmas, which will be used in the following proofs. Lemma 3.1 is from He and Xie (in press). Lemma 3.3 is from Fuk and Nagaev (1971). Φ (x) is the standard normal distribution function, Φ (x) = √1
2π
φ(x) = Φ ′ (x).
x −∞
e−
t2 2
dt,
Lemma 3.1. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. standard normal distribution random variables. Then
ϵ 2 λX (ϵ) = ϵ 2
∞
2
√
2π
n =1
ϵ2 2 e−t /2 dt = 1 − + O(ϵ 3 ), 2 ϵ n ∞
as ϵ → 0.
√
If {Xn , n ≥ 1} is a sequence of independent random variables with mean zero and finite variance. Put EXj2 = σj2 , Bn =
σj2 . Bikelis (1966) proved the following inequality: √ n n (1+|x|) Bn 1 −3/2 −3 Xj < x − Φ (x) ≤ CBn (1 + |x|) u2 dVj (u)dv P √ Bn j=1 |u|>v j =1 0
n
j =1
for every x, where Vj (x) = P (Xj < x) is the distribution function of random variable Xj . By applying this inequality to a √ sequence of i.i.d. random variables with mean zero and variance 1, let x = ϵ n, we get the following. Lemma 3.2. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with mean zero and EX 2 = 1. Then for any ϵ > 0,
∞ (1+ϵ √n)√n C −t 2 / 2 P (|Sn | > nϵ) − √2 e dt ≤ TX (v)dv. √ √ √ ( 1 + ϵ n) 3 n 0 2π ϵ n Lemma 3.3. Let {X , Xn , n ≥ 1} be a sequence of i.i.d. random variables with E |X |β < ∞, where 1 < β ≤ 2. For x > 0, y > 0, we have P (|Sn | > x) ≤ nP (|X | ≥ y) + 2nx/y
eE |X |β
x/y
xyβ−1
.
Proof of Theorem 2.1. We prove Theorem 2.1 in two cases: σ 2 = 1, and σ 2 ̸= 1. (a) σ 2 = 1, assume that 0 < ϵ < 1, note that ∞ 2ϵ 2 ∞ −t 2 /2 ϵ 2 λX (ϵ) = I + √ e dt , √ 2π n=1 ϵ n
where I = ϵ2
∞ n=1
2 P (|Sn | > nϵ) − √ 2π
∞
2 e−t /2 dt √
ϵ n
.
Applying Lemma 3.1, we get
ϵ 2 λX (ϵ) = I + 1 −
ϵ2 2
+ O(ϵ 3 ),
as ϵ → 0.
(3.1)
By (3.1), in order to prove Theorem 2.1, we only need to show that
I = O ϵ 1/2 + TX (ϵ −1/2 ) log
1 TX (ϵ −1/2 ) + ϵ 1/2
,
as ϵ → 0.
We write
I = ϵ 2
K
ϵ2
n=1
+ϵ
2
∞ 2 P (|Sn | > nϵ) − √
n= K2 +1 ϵ
2π
∞ √ ϵ n
e
−t 2 /2
dt
=: I1 + I2 ,
T. Xie, J. He / Statistics and Probability Letters 82 (2012) 1576–1582
1579
where [x] denotes the integer part of x, and K is a large enough positive integer which will be determined later. First, we shall deal with the term I1 , by Lemma 3.2,
| I1 | ≤ ϵ
2
K
ϵ2
√ √ (1 + ϵ n)3 n
n=1
3/4
1
ϵ2
2 = ϵ
+ϵ
√ √ (1+ϵ n) n
C
1
ϵ2
2
n=1
n=
1 ϵ2
TX (t )dt
0
+ϵ
3/4
2 n=
+1
K
(1+ϵ ϵ2 C √ √ (1 + ϵ n)3 n 0 1 ϵ2
√ √ n) n
TX (t )dt
+1
= : I11 + I12 + I13 . Since TX (t ) ≤ EX 2 = 1, and TX (t ) is a nonnegative decreasing function, then
3/4
1
ϵ2
C ≤ C ϵ 1/2 = O(ϵ 1/2 ).
I11 ≤ ϵ 2
(3.2)
n =1
For I12 , noting that ( ϵ12 )3/4 < n < ϵ12 implies √1ϵ < nϵ < 1ϵ , we can obtain √1ϵ < (1 + ϵ
I12 ≤ ϵ
2 n=
1 ϵ2
≤ϵ
1
ϵ2
n=
C
1
√ √ +ϵ n ϵ
dt
ϵ2
2
+1
1
n=
1 ϵ2
√1 ϵ
+1
3/4
1
√ ϵ
0
1 ϵ2
1 ϵ2
dt + TX
n
2
√
3/4
√ √ (1+ϵ n) n
√
1/ ϵ
C
√ √ n) n ≤ 2ϵ . Thus,
3/4
C
√ TX ϵ n
1
√
ϵ
+1
1 = O ϵ 2 + TX (ϵ −1/2 ) .
(3.3)
Analogously,
I13 ≤ ϵ
K
ϵ2
n=
1 ϵ2
K
3
n=1+
≤ C ϵ
1
1
ϵ 3 n2
+ C ϵ 2 TX
1
√
ϵ
K
1 ϵ2
1
ϵ 3 n2
+ TX
1
√ ϵ
dt
√
1/ ϵ
1
ϵ2
√ (1 + ϵ n)2
1
n=1+
√ √ (1+ϵ n) n
1
K
√ ϵ
ϵ2
ϵ2
n=1+
n=1+
1
K
ϵ2
ϵ2
3 2
0
ϵ2
≤ Cϵ 2
dt + TX
√ √ ( 1 + ϵ n) 3 n
+1
√
1/ ϵ
C
2
1 ϵ2
n
1 = O ϵ 2 + TX (ϵ −1/2 ) log K .
(3.4)
From (3.2) to (3.4), we get I1 = O ϵ 1/2 + TX (ϵ −1/2 ) log K .
(3.5)
Next, we estimate the term I2 , using the following inequalities 1 − Φ (x) ≤ 2
√
2π
√
φ(x)
√ 2φ(ϵ n) 2 e−t /2 dt = 2(1 − Φ (ϵ n)) ≤ = O(ϵ −5 n−5/2 ). √ √ ϵ n ϵ n
∞
x
, x ∈ R+ and t 2 e−t ≤ 1, t > 2. We can obtain
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So, I2 = ϵ
∞
2
P (|Sn | > nϵ) − ϵ
∞
∞ n=
K ϵ2
2π
n= K2 +1 ϵ
= ϵ2
√
∞
P (|Sn | > nϵ) + C ϵ 2
∞
2
n= K2 +1 ϵ
n= K2 +1 ϵ
≤ ϵ2
∞
2
√
2 e−t /2 dt
ϵ n
ϵ −5 n−5/2
n= K2 +1 ϵ
P (|Sn | > nϵ) + O(K −3/2 ).
(3.6)
+1
Using the method of Heyde (1975) to estimate the first term of (3.6), and specializing Lemma 3.3 by x = 2y = nϵ and β = 2, we have P (|Sn | > nϵ) ≤ nP
|X | ≥
1 2
nϵ
+ 8e2 n−2 ϵ −4 .
Hence ∞
I2 ≤ ϵ 2
nP
1
|X | >
2
n= K2 +1 ϵ
∞
= ϵ2 n=
K ϵ2
n +1
∞
= ϵ2
|x|≥ 12 nϵ
n
n= K2 +1 ϵ
∞
= ϵ2
k= K2 +1 ϵ
∞
≤C k=
K ϵ2
dFX (x) + O(K −1 )
1 kϵ≤|x|< 12 (k+1)ϵ 2
dFX (x) + O(K −1 )
dFX (x)
k
1 kϵ≤|x|< 12 (k+1)ϵ 2
dFX (x) + O(K −1 )
≤C |x|≥ 2Kϵ
n + O(K −1 )
n=1
+1
ϵ −4 n−2 + O(K −3/2 )
n= K2 +1 ϵ
1 kϵ≤|x|< 12 (k+1)ϵ 2
(kϵ)2
∞
+ 8e2 ϵ 2
∞ k=n
nϵ
x2 dFX (x) + O(K −1 ) = O TX
K
2ϵ
+ K −1 .
(3.7)
Together (3.5) with (3.7), we get
I = O ϵ 1/2 + K −1 + TX (ϵ −1/2 ) log K + TX
K
2ϵ
.
Set K = [ T (ϵ −1/12 )+ϵ 1/2 ], then X K lim √ = +∞,
ϵ→0
ϵ
TX (t ) is a nonnegative decreasing function, then for sufficiently small ϵ > 0, we have
TX
K 2ϵ
= TX
K
√ ϵ −1/2 2 ϵ
≤ TX (ϵ −1/2 ).
Therefore
I = O ϵ 1/2 + TX (ϵ −1/2 ) log
1 TX (ϵ −1/2 ) + ϵ 1/2
,
as ϵ → 0.
Then
ϵ 2 λX (ϵ) = 1 + O ϵ 1/2 + TX (ϵ −1/2 ) log
1 TX (ϵ −1/2 ) + ϵ 1/2
,
as ϵ → 0.
T. Xie, J. He / Statistics and Probability Letters 82 (2012) 1576–1582
1581
(b) σ 2 ̸= 1, then
∞ ϵ 2 ϵ |Sn | ϵ λX (ϵ) − σ = σ P −1 ≥n σ n=1 σ σ = σ 2 O σ −1/2 ϵ 1/2 + TX /σ (σ 1/2 ϵ −1/2 ) log 2
2
2
= O σ 3/2 ϵ −1/2 + TX (σ 3/2 ϵ −1/2 ) log Thus, the proof of Theorem 2.1 is now complete.
1 TX /σ (σ 1/2 ϵ −1/2 ) + σ −1/2 ϵ 1/2
σ2 3 / 2 − 1 / TX (σ ϵ 2 ) + σ 3/2 ϵ 1/2
.
Proof of Theorem 2.2. Without the loss of generality, we assume that σ 2 = 1, 0 < ϵ < proof of Theorem 2.1, we have
ϵ2 ϵ 2 λX (ϵ) − 1 = I − + O(ϵ 3 ) 2 1 3 ∞ ϵ 2 2 2 = ϵ +ϵ P (|Sn | > nϵ) − √ n=1
n=
1
ϵ3
2π
+1
∞
√ ϵ n
2
e
− t2
dt
1 . 2
By the process similar to the
+ O(ϵ 2 )
= : I1 + I2 + O(ϵ 2 ). First, we estimate the term I2 . It is easily seen that ∞
I2 ≤ ϵ 2 n=
1
ϵ3
nP
n=
1
ϵ3
∞
+ 8e2 ϵ 2 n=
|x|≥ 12 nϵ
+1
1 2ϵ 2
+ϵ
1
ϵ3
∞
ϵ − 4 n− 2 + C ϵ 2 +1
n=
1
ϵ3
ϵ −5 n−3/2 +1
dFX (x) + O(ϵ) + O(ϵ 3/2 )
n
= O TX
2
nϵ
+1
∞
= ϵ2
1
|X | >
1
= O TX
ϵ
+ϵ
= O(ϵ δ + ϵ).
(3.8)
For I1 , we have
1
ϵ3
C
√ √ (1+ϵ n) n
TX (t )dt √ √ (1 + ϵ n)3 n 0 1 1 (1+ϵ √n)√n ϵ2 ϵ3 C 2 2 = ϵ +ϵ TX (t )dt √ √ (1 + ϵ n)3 n 0 n =1 1
| I1 | ≤ ϵ
2
n=1
n=
+1
ϵ2
= : I11 + I12 .
(3.9)
For I11 , since TX (t ) ≤ EX 2 = 1, TX (t ) = O(t −δ ), and noting that n ≤ ϵ12
I11 = ϵ 2
1
ϵ2
n =1
√ √ (1+ϵ n) n
C
√
( 1 + ϵ n)
√ 3
n
√ √ √ implies (1 + ϵ n) n ≤ 2 n ≤ 2ϵ , we can get
TX (t )dt
0
1 ϵ2 C 2 ≤ϵ √ 1+
n =1
n
≤ Cϵ 1 +
2/ϵ
TX (t ) dt
1 2/ϵ
t
−δ
dt
1
1 = O(ϵ δ + ϵ)I (δ ̸= 1) + O ϵ log + 1 I (δ = 1) . ϵ
(3.10)
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T. Xie, J. He / Statistics and Probability Letters 82 (2012) 1576–1582
For I12 , noting that ϵ12 ≤ n ≤ ϵ13 implies ϵ
I12 = ϵ
ϵ3
2 n=
1 ϵ2
1
+1
n=
√ √ (1 + ϵ n)3 n
1
+1
√ (1 + ϵ n)3 n
ϵ3
2 n=
1 ϵ2
2nϵ
√
1 ϵ2
C +1
n2 ϵ 3
1+
TX (t )dt
0
2nϵ
TX (t )dt
0
C
n > 1 and nϵ ≤ ϵ12 , it follows
√ √ (1+ϵ n) n
C
ϵ3
≤ ϵ2
≤ϵ
1
√
t
−δ
dt
1
1
δ
= O(ϵ + ϵ)I (δ ̸= 1) + O ϵ log
ϵ
+ 1 I (δ = 1) .
(3.11)
By (3.8)–(3.11), we obtain that
O(ϵ δ ), 1 2 ϵ λX (ϵ) − 1 = O ϵ log +1 , ϵ O(ϵ),
0 < δ < 1,
δ=1 δ > 1.
If σ 2 ̸= 1, note that λX (ϵ) = λ X ( σϵ ), we have σ
ϵ 2 λX (ϵ) − σ 2 = σ 2
ϵ 2 σ
λX σ
ϵ σ
2−α α ϵ ), O(σ σ 2 − σ = O σ ϵ log +1 , ϵ O(σ ϵ),
The proof of Theorem 2.2 is now complete.
0 < δ < 1,
δ = 1, δ > 1.
Acknowledgments The authors are grateful to anonymous referees and the editor for their constructive comments and suggestions, which have improved the presentation of the paper. References Bikelis, A., 1966. Estimates of the remainder in the central limit theorem. Litovsk. Math. Sb. 6, 323–346. Erdös, P., 1949. On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 286–291. Erdös, P., 1950. Remark on my paper on a theorem of Hsu and Robbins. Ann. Math. Statist. 21, 138. Fuk, D.H., Nagaev, S.V., 1971. Probability inequalities for sums of independent random variables. Theory Probab. Appl. 16, 643–660. Gut, A., Spˇataru, A., 2000. Precise asymptotics in the Baum–Katz and Davis law of large numbers. J. Math. Anal. Appl. 248, 233–246. Gut, A., Steinebach, J., 2012. Convergence rates in precise asymptotics. J. Math. Anal. Appl. 390, 1–14. Hsu, P.L., Robbins, H., 1947. Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. USA 33, 25–31. Heyde, C.C., 1975. A supplement to the strong law of large numbers. J. Appl. Probab. 12, 903–907. He, J.J., Xie, T.F., 2012. Asymptotic property for some series of probability, Acta Math. Appl. Sin. http://dx.doi.org/10.1007/s10255-012-0138-6 (in press). Jiang, Y., Zhang, L.X., 2007. Precise rates in the law of logarithm for the moment convergence of i.i.d. random variables. J. Math. Anal. Appl. 327, 695–714. Klesov, O.I., 1994. On the convergence rate in a theorem of Heyde. Theory Probab. Math. Statist. 49, 83–87. Liu, W.D., Lin, Z.Y., 2006. Precise asymptotics for a new kind of complete moment convergence. Statist. Probab. Lett. 76, 1787–1799.