Precise asymptotics for the linear processes generated by associated random variables in Hilbert spaces

Computers and Mathematics with Applications 64 (2012) 1937–1943

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Precise asymptotics for the linear processes generated by associated random variables in Hilbert spaces✩, ✩✩ Ke-Ang Fu a,∗ , Jie Li b , Ya-Juan Dong a , Hui Zhou c a

School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China

b

School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou 310018, China

c

College of Economics, Hangzhou Dianzi University, Hangzhou 310018, China

article

abstract

info

Article history: Received 6 August 2010 Received in revised form 12 November 2011 Accepted 15 March 2012

Let {εk , k ∈ Z} be a strictly stationary associated sequence of random variables taking values in a real separable Hilbert space, ∞and {ak ; k ∈ Z} be a sequence of bounded linear operators. For a linear process Xk = i=−∞ ai (εk−i ), the precise probability and moment n convergence rates of i=1 Xi in some limit theorems are discussed. © 2012 Elsevier Ltd. All rights reserved.

Keywords: Association Bounded operator Convergence rates Hilbert space Linear processes

1. Introduction and main results Let H be a separable real Hilbert space with the norm ∥ · ∥ generated by an inner product, ⟨·, ·⟩H and let {ei ; i ≥ 1} be an orthonormal basis in H. Let L(H) be the class of bounded linear operators from H to H and denote by ∥·∥L(H) its usual uniform norm. Let {εk , k ∈ Z} be a sequence of H-valued random variables, and {ak , k ∈ Z} be a sequence of operators, ak ∈ L(H). Define the stationary Hilbert space process by Xk =

∞ 

ai (εk−i ),

k ∈ Z,

(1.1)

i=−∞

provided the series is convergent in some sense. The sequence {Xk , k ∈ Z} is a natural extension of the multivariate linear processes [1]. These types of processes with values in functional spaces also facilitate the study of estimation and forecasting problems for several classes of continuous time processes, and one can refer [2] for more details. It is noted that when {εk , k ∈ Z} is a strong H-white noise (i.e. a sequence of i.i.d. H-valued random variables such that n 0 < E∥εk ∥2 < ∞ and Eεk = 0), the series in (1.1) converges almost surely and in L1 (H), and Sn = i=1 Xi satisfies the ∞ central limit theorem, provided i=−∞ ∥ai ∥L(H) < ∞ [3,4]. Moreover, Bosq [5] established a Berry–Esseen type inequality ∞ with an additional condition i=1 i∥ai ∥L(H) < ∞.

✩ Project supported by Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LQ12A01018 and Q12A010066) and Department of Education of Zhejiang Province (Grant No. Y201119891). ✩✩ The paper has been evaluated according to old Aims and Scope of the journal.

∗

Corresponding author. E-mail address: [email protected] (K.-A. Fu).

0898-1221/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2012.03.046

1938

K.-A. Fu et al. / Computers and Mathematics with Applications 64 (2012) 1937–1943

Recently, some researchers have investigated the limit theorems of the linear process Xk by assuming that {εk , k ∈ Z} is a strictly stationary sequence of (negatively) associated H-valued random variables, which extend many previous results. For example, Ko and Kim [6] studied the functional central limit theorem. Before stating their results, we first introduce the notions of associated random variables, associated random vectors and H-valued associated random variables (See [7,6,8], respectively). Definition 1.1. A finite sequence of real-valued random variables {Xk ; 1 ≤ k ≤ n} is said to be associated, if Cov{f (X1 , . . . , Xn ), g (X1 , . . . , Xn )} ≥ 0,

whenever f and g are coordinatewise increasing and the covariance exists. An infinite sequence of random variables is associated if every finite subsequence is associated. Definition 1.2. A finite sequence of Rd -valued random vectors {Xk ; 1 ≤ k ≤ n} is said to be associated, if for all coordinatewise increasing functions f , g : Rnd → R Cov{f (X1 , . . . , Xn ), g (X1 , . . . , Xn )} ≥ 0,

whenever the covariance exists. An infinite sequence of random vectors is associated if every finite subsequence is associated. Definition 1.3. A sequence of H-valued random variables {Xk ; k ≥ 1} is said to be associated, if for some orthonormal basis {ei ; i ≥ 1} in H and for any d ≥ 1, the d-dimensional sequence (⟨Xi , e1 ⟩H , . . . , ⟨Xi , ed ⟩H ), i ≥ 1, is associated. The main result of Ko and Kim [6] reads as follows. Theorem  A. Let Xk be an H-valued linear processes given by (1.1), where {ak , k ∈ Z} is a sequence of linear bounded operator ∞ satisfying stationary associated sequence of H-valued random variables with i=−∞ ∥ai ∥L(H) < ∞, and {εk ; k ∈ Z} is a strictly ∞ Eε1 = 0 and 0 < E∥ε1 ∥2 < ∞. If τ 2 := E∥ε1 ∥2 + 2 i=2 E(⟨ε1 , εi ⟩H ) < ∞, then we have n−1/2

[nt ] 

Xi → W

in distribution,

i=1

where W is a Wiener process on H with covariance operator AΓ A∗ , A = k, ℓ = 1, 2, . . ., and

τkℓ = E(⟨ek , ε1 ⟩H ⟨eℓ , ε1 ⟩H ) +

∞

i=−∞

ai , A∗ is the adjoint operator of A, Γ = (τkℓ ),

∞  [E(⟨ek , ε1 ⟩H ⟨eℓ , εi ⟩H ) + E(⟨eℓ , ε1 ⟩H ⟨ek , εi ⟩H )]. i=2

Inspired by them, in this paper we aim to further study the limit properties of linear processes generated by dependent H-valued random variables, and the exact probability and moment convergence rates of Sn in some limit theorems are derived. Let {εk ; k ∈ Z} be a strictly stationary sequence of associated H-valued random variables. Let G be an H-valued Gaussian random variable with mean zero and covariance AΓ A∗ . Denote the largest eigenvalue of AΓ A∗ by σ 2 . Let l be the dimension of the corresponding eigenspace, and let σi2 , 1 ≤ i ≤ l′ be the positive eigenvalues of AΓ A∗ arranged in a nonincreasing order and take into account the multiplicities. Further, if l′ < ∞, put σi2 = 0, i ≥ l′ . Note that we always have σi2 = σ 2 , 1 ≤ i ≤ l and σi2 < σ 2 , i > l [9]. Now it is in a position to state our main results. Theorem 1.1. Let Xk be an H-valued linear processes given by (1.1), where {ak , k ∈ Z} and {εk ; k ∈ Z} are defined as Theorem A. Then under the assumptions of Theorem A, we have that for any δ > −1, lim ϵ 2(δ+1)

ϵ↘0

n =1

lim ϵ 2(δ+1)

ϵ↘0

∞  (log log n)δ

n log n

∞  (n log n)δ n =1

n

P(∥Sn ∥ ≥ ϵσ

P(∥Sn ∥ ≥ ϵσ





2n log log n) =

n log n) =

E∥G∥2(δ+1)

(δ + 1)(2σ 2 )(δ+1)

E∥G∥2(δ+1)

(δ + 1)(2σ 2 )δ+1

,

(1.2)

,

Theorem 1.2. Under the conditions of Theorem 1.1, we have that for any δ > −1/2 lim ϵ 2δ+1

ϵ↘0

lim ϵ 2δ+1

ϵ↘0

∞  (log log n)δ−1/2 n=1

n3/2 log n

∞  (log n)δ−1/2 n=1

n3/2

E{∥Sn ∥ − ϵσ

E{∥Sn ∥ − ϵσ





2n log log n}+ =

n log n}+ =

E∥G∥2(δ+1)

(δ + 1)(2δ + 1)(2σ 2 )δ+1/2 E∥G∥2(δ+1)

(δ + 1)(2δ + 1)(2σ 2 )δ+1/2

,

,

(1.3)

K.-A. Fu et al. / Computers and Mathematics with Applications 64 (2012) 1937–1943

1939

where {x}+ = max(x, 0). Theorem 1.3. Let 1 < p < 2 and r > 1 + p/2. Under the conditions of Theorem 1.1, we have ∞ 

lim ϵ 2(r −p)/(2−p)

ϵ↘0

nr /p−2 P(∥Sn ∥ ≥ ϵ n1/p ) =

n =1

lim ϵ 2(r −p)/(2−p)−1

ϵ↘0

∞ 

p r −p

E∥G∥2(r −p)/(2−p) ,

nr /p−2−1/p E{∥Sn ∥ − ϵ n1/p }+ =

n=1

p(2 − p) E∥G∥2(r −p)/(2−p) . (r − p)(2r − p − 2)

Remark 1.1. We extend the results of Li [10,11] for real-valued linear process to Hilbert setting, and weaken the moment assumption of ε1 by avoiding the application of the Berry–Esseen bound. As an application of Theorem 1.3, we have the following complete moment convergence and complete convergence. Corollary 1.1. Let 1 < p < 2 r > 1 + p/2. Then under the conditions of Theorem 1.1, we have that for any ϵ > 0, ∞ 

nr /p−2−1/p E{∥Sn ∥ − ϵ n1/p }+ < ∞

n =1

∞ 

nr /p−2 P(∥Sn ∥ > ϵ n1/p ) < ∞.

n =1

2. Proofs of Theorems 1.1 and 1.2 From this section on, we begin to prove the theorems, and in the sequel, let C denote positive constants whose values p

d

possibly vary from place to place. The notation an ∼ bn means that ban → 1 as n → ∞, → (→) denotes convergence in n distribution (probability) and [x] denotes the greatest integer part of x. Since the proof is similar, we only give the proof of (1.2) and (1.3) here, and the proof of others are omitted. Now we state two lemmas which will be used later. Lemma 2.1. Let G be a nondegenerate Gaussian random variables with mean zero and covariance operator AΓ A∗ . Then for y > 0, P(∥G∥ ≥ y) ∼ 2Dσ 2 yl−2 exp

 −

y2



2σ 2

,

as y → ∞,

l

where D = (2σ 2 )− 2 Γ −1 ( 2l )K (AΓ A∗ ) and K (AΓ A∗ ) =

∞

i=l+1

((σ 2 − σi2 )/σ 2 )−1/2 .

Lemma 2.2. For n ≥ 1, let αn (ϵ) > 0, βn (ϵ) > 0 and f (ϵ) > 0 satisfying

αn (ϵ) ∼ βn (ϵ),

as n → ∞ and ϵ → δ0 ,

and f (ϵ)βn (ϵ) → 0,

as ϵ → δ0 , ∀n ≥ 1.

Then





lim sup lim inf f (ϵ) ϵ→δ0

ϵ→δ0

∞ 





αn (ϵ) = lim sup lim inf f (ϵ)

n=1

ϵ→δ0

ϵ→δ0

Proof. See Lemmas 2.1 and 2.4 of Huang and Zhang [12].

∞ 

βn (ϵ).

n =1



∞

Note that E{|X | − n}+ = n P(|X | ≥ x)dx, which shows the relationship between Theorems 1.1 and 1.2, and thus we put emphasis on the proof of (1.3). Proposition 2.1. For any δ > −1/2, we have lim ϵ 2δ+1

ϵ↘0

∞  (log log n)δ−1/2 n =1

n log n

E{∥G∥ − ϵσ



2 log log n}+ =

E∥G∥2(δ+1)

(δ + 1)(2δ + 1)(2σ 2 )1/2+δ

,

(2.1)

1940

K.-A. Fu et al. / Computers and Mathematics with Applications 64 (2012) 1937–1943

and for any δ > −1, we have lim ϵ 2(δ+1)

∞  (log log n)δ

ϵ↘0

n log n

n =1

P(∥G∥ ≥ ϵσ

E∥G∥2(δ+1)

2 log log n) =



(δ + 1)(2σ 2 )(δ+1)

.

(2.2)

Proof. Note that lim ϵ 2δ+1

∞  (log log n)δ−1/2

ϵ↘0

n=1

= lim ϵ 2δ+1

(log log y)δ−1/2

∞



ϵ↘0

∞



P{∥G∥ ≥ x}dxdy

∞



P{∥G∥ ≥ x}dxdz x



lim P{∥G∥ ϵ↘0 √2σ ϵ 2(δ+1)

(2σ 2 )−1/2−δ E∥G∥ (δ + 1)(2δ + 1)

∞

P{∥G∥ ≥ x} √

ϵ↘0 √2σ ϵ  ∞ 2 −1/2−δ

2δ + 1

2 log log n}+

z

= 2(2σ 2 )−1/2−δ lim 2(2σ )



√ ϵσ 2 log log y

z 2δ

ϵ↘0 √2σ ϵ  ∞

=



y log y

ee

= 2(2σ 2 )−1/2−δ lim

=

E{∥G∥ − ϵσ

n log n

z 2δ dzdx

2σ ϵ

≥ x}x2δ+1 dx

,

which means (2.1) is valid. Similarly, by some calculations, we have (2.2) holds true.



In what follows, set c (ϵ, M ) = exp(exp(M /ϵ 2 )) and assume that σ = 1 without loss of generality. Proposition 2.2. Under the assumptions of Theorem 1.1, we have that for any M > 4 and δ > −1/2, lim ϵ 2δ+1

(log log n)δ−1/2



ϵ↘0

n log n

n≤c (ϵ,M )

  |n−1/2 E{∥Sn ∥ − ϵ 2n log log n}+ − E{∥G∥ − ϵ 2 log log n}+ | = 0,

(2.3)

and for any δ > −1, we have lim ϵ 2(δ+1)

ϵ↘0

(log log n)δ

 n≤c (ϵ,M )

n log n

  |P(∥Sn ∥ ≥ ϵσ 2n log log n) − P(∥G∥ ≥ ϵσ 2 log log n)| = 0.

√

(2.4)

√

d

Proof. Denote ∆n = supx |P(∥Sn / n∥ ≥ x)− P(∥G∥ ≥ x)|. It is readily seen that Theorem A implies Sn / n → G as n → ∞, which leads to ∆n → 0 as n → ∞. Note that

ϵ 2δ+1

(log log n)δ−1/2



n log n

n≤c (ϵ,M )

=ϵ

2δ+1

 n≤c (ϵ,M )

∞



  |n−1/2 E{∥Sn ∥ − ϵ 2n log log n}+ − E{∥G∥ − ϵ 2 log log n}+ |

  (log log n)δ−1/2  −1/2 n  n log n

∞

P(∥Sn ∥ ≥ x + ϵ

2n log log n)dx



0

  P(∥G∥ ≥ x + ϵ 2 log log n)dx  (log log n)δ  Γn   |P(∥Sn ∥ ≥ (x + ϵ) 2n log log n) − P(∥G∥ ≥ (x + ϵ) 2 log log n)|dx 

− 0

≤ C ϵ 2δ+1

n≤c (ϵ,M )

+ C ϵ 2δ+1

 n≤c (ϵ,M )

+ C ϵ 2δ+1

 n≤c (ϵ,M )

n log n

0

δ



δ



(log log n) n log n

=: C ((I ) + (II ) + (III )),

P(∥Sn ∥ ≥ (x + ϵ) 2n log log n)dx



Γn

(log log n) n log n

∞

∞

P(∥G∥ ≥ (x + ϵ) 2 log log n)dx



Γn

K.-A. Fu et al. / Computers and Mathematics with Applications 64 (2012) 1937–1943

−1/2

where Γn = (log log n)−1/2 ∆n m

1



(log log m)δ+1/2

n =1

1/2 ∆n

1941

. Then an application of the Toeplitz Lemma leads to

(log log n)δ−1/2

→ 0,

n log n

as m → ∞.

Hence,

(I ) ≤ ϵ 2δ+1

1/2

∆n (log log n)δ−1/2



n log n

n≤c (ϵ,M )

= ϵ 2δ+1 (log log[c (ϵ, M )])δ+1/2 ≤ M δ+1/2

1



(log log[c (ϵ, M )])δ+1/2

n≤c (ϵ,M )

1/2

∆n (log log n)δ−1/2 n log n

1/2

∆n (log log n)δ−1/2

1



(log log[c (ϵ, M )])δ+1/2

n≤c (ϵ,M )

n log n

→ 0,

as ϵ ↘ 0.

As to (III ), it follows



∞

P(∥G∥ ≥ (x + ϵ) 2 log log n)dx ≤ C (log log n)



Γn

−1



∞

Γn

(x + ϵ)−2 dx

≤ C (log log n)−1/2 ∆n 1/2 .

(2.5)

Then using the Toeplitz Lemma again entails

(III ) ≤ C ϵ 2δ+1

(log log n)δ−1/2



n log n

n≤c (ϵ,M )

∆1n/2 → 0 as ϵ → 0.

As for (II ), it follows that for any y > 0, as n → ∞,

√

P(∥Sn /

n∥ ≥ y) ∼ P(∥G∥ ≥ y),

and also recall that P(∥G∥ > y) ∼ 2Dσ 2 yl−2 exp

y2

 −



2σ 2

,

as y → ∞.

Thus an application of the Lemma 2.2 entails lim sup ϵ 2δ+1 ϵ↘0

(log log n)δ



n log n

n≤c (ϵ,M )

≤ lim sup ϵ 2δ+1 ϵ↘0

∞ 

= lim sup ϵ 2δ+1 ϵ↘0

= lim sup ϵ 2δ+1 ϵ↘0



∞



n log n





n log n

≤ C lim sup ϵ 2δ+1 ϵ↘0

∞  n =1

≤ C lim sup ϵ 2δ+1 ϵ↘0

∞

P(∥G∥ ≥ (x + ϵ) 2 log log n)dx



Γn

 ∞  (log log n)δ n=1

P(∥Sn ∥ ≥ (x + ϵ) 2n log log n)dx

Γn

δ

(log log n)

n=1

P(∥Sn ∥ ≥ (x + ϵ) 2n log log n)dx

Γn

(log log n)δ

n =1

∞ 

∞



n log n

∞



Γn

δ−1

(log log n)

2Dσ 2 ((x + ϵ) 2 log log n)l−2 exp(−((x + ϵ) 2 log log n)2 /2σ 2 )dx



n log n

n =1

∞ Γn

∞  (log log n)δ−1/2

n log n



(x + ϵ)−2 dx

∆1n/2 ,

(2.6)

where we use the inequality e−x ≤ x−Q for any x, Q > 0. This, combined with the Toeplitz Lemma, leads to limϵ↘0 (II ) = 0. Hence, the proof of (2.3) is complete. As for (2.4), applying ∆n → 0 as n → ∞ with the Toeplitz Lemma, of course ensures (2.4) holds.  Proposition 2.3. For 0 < ϵ < 1/4, we have uniformly lim ϵ 2δ+1

M →∞

 n>c (ϵ,M )

(log log n)δ−1/2 n log n

E{∥G∥ − ϵ



2 log log n}+ = 0,

(2.7)

1942

K.-A. Fu et al. / Computers and Mathematics with Applications 64 (2012) 1937–1943

and lim ϵ 2(δ+1)

M →∞

(log log n)δ



P(∥G∥ ≥ ϵ

n log n

n>c (ϵ,M )

2 log log n) = 0.



(2.8)

Proof. Note that for k large enough,

ϵ 2δ+1

(log log n)δ−1/2



n log n

n>c (ϵ,M )

≤ C ϵ 2δ+1

∞



P{∥G∥ ≥ (x + ϵ) 2 log log n}dx



∞



n log n

n>c (ϵ,M )

(x + ϵ) (log log n)k/2

0

dx

ϵ −k+1

n log n

n>c (ϵ,M )

E∥G∥k k

(log log n)δ−k/2



2 log log n + x}dx

0

(log log n)δ





0

n log n

n>c (ϵ,M )

≤ C ϵ 2δ+1

P{∥G∥ ≥ ϵ

(log log n)δ



≤ C ϵ 2δ+1

∞



= CM δ+1−k/2 → 0, when M → ∞, uniformly for 0 < ϵ < 1/4, which ends the proof of (2.7). Similarly, (2.8) is also valid.



Proposition 2.4. Under the assumptions of Theorem 1.1, we have that for any δ > −1/2, lim lim ϵ 2δ+1

M →∞ ϵ↘0

(log log n)δ−1/2



n3/2

n>c (ϵ,M )

log n

E{∥Sn ∥ − ϵ



2 log log n}+ = 0,

(2.9)

and lim lim ϵ 2(δ+1)

M →∞ ϵ↘0

(log log n)δ



n log n

n>c (ϵ,M )

P(∥Sn ∥ ≥ ϵ

2n log log n) = 0.



(2.10)

Proof. Notice that for any ε > 0 and any x > 0, there exists an n0 such that for n ≥ n0 ,

  √  P(∥Sn / n∥ ≥ x)     P(∥G∥ ≥ x)  ≤ (1 + ε). Also recall that Lemma 2.1 entails that P(∥G∥ ≥ x) ≤ Cxl−2 e−x /2 for x large enough. Then for ϵ sufficiently small and M sufficiently large, it yields 2

ϵ 2δ+1

(log log n)δ−1/2



n3/2 log n

n>c (ϵ,M )

≤ Cϵ

2δ+1

(log log n)δ



≤ C ϵ 2δ+1 ≤ Cϵ

2δ+1

2



∞

y log y

s M

≤ C ϵ2

t δ−1/2

M∞ 

∞

P{∥Sn ∥ ≥ (x + ϵ) 2n log log n}dx



∞



 ((x + ϵ) log log y)l−2 exp(−(x + ϵ)2 log log y/2)dxdy 0

(log log y)δ−1/2

∞



y log y

c (ϵ,M )

≤ C ϵ2 ≤ Cϵ

(log log y)

c (ϵ,M )





−1 −s

t

2n log log n}+

0

δ

∞



∞



n log n

n>c (ϵ,M )



E{∥Sn ∥ − ϵ

∞

s−1 e−s dsdy ϵ 2 log log y

∞

s−1 e−s dsdt s

e

t δ−1/2 dtds

M

sδ−1/2 e−s ds.

M

Hence, (2.9) follows, as desired. Also from the proof process, we can get (2.10) similarly.



K.-A. Fu et al. / Computers and Mathematics with Applications 64 (2012) 1937–1943

1943

Proofs of (1.2) and (1.3). From Propositions 2.1–2.4, (1.2) and (1.3) follows immediately by using the triangle inequality.  3. Proof of Theorem 1.3 Similarly as above, we can get Theorem 1.3 by applying the following propositions, and the proofs are omitted. Proposition 3.1. For 1 < p < 2 and r > 1 + p/2, we have lim ϵ (2r −p−2)/(2−p)

ϵ↘0

∞ 

nr /p−3/2−1/p E{∥G∥ − ϵ n1/p−1/2 }+ =

n=1

p(2 − p) E∥G∥2(r −p)/(2−p) , (r − p)(2r − p − 2)

and lim ϵ 2(r −p)/(2−p)

ϵ↘0

∞ 

nr /p−2 P(∥G∥ ≥ ϵ n1/p−1/2 ) =

n =1

p r −p

E∥G∥2(r −p)/(2−p) .

Proposition 3.2. Set a(ϵ, M ) = [M ϵ −2p/(2−p) ]. Then for 1 < p < 2 and r > 1 + p/2, lim ϵ (2r −p−2)/(2−p)

ϵ↘0

nr /p−2−1/p |n1/2 E{∥G∥ − ϵ n1/p−1/2 }+ − E{∥Sn ∥ − ϵ n1/p }+ | = 0,

 n≤a(ϵ,M )

and lim ϵ 2(r −p)/(2−p)

ϵ↘0

nr /p−2 |P(∥G∥ ≥ ϵ n1/p−1/2 ) − P(∥Sn ∥ ≥ ϵ n1/p )| = 0.

 n≤a(ϵ,M )

Proposition 3.3. For 1 < p < 2 and r > 1 + p/2, we have lim lim ϵ (2r −p−2)/(2−p)

M →∞ ϵ↘0

nr /p−3/2−1/p E{∥G∥ − ϵ n1/p−1/2 }+ = 0,

 n>a(ϵ,M )

and lim lim ϵ 2(r −p)/(2−p)

M →∞ ϵ↘0



nr /p−2 P(∥G∥ ≥ ϵ n1/p−1/2 ) = 0.

n>a(ϵ,M )

Proposition 3.4. For 1 < p < 2 and r > 1 + p/2, we have lim lim ϵ (2r −p−2)/(2−p)

M →∞ ϵ↘0

nr /p−2−1/p E{∥Sn ∥ − ϵ n1/p }+ = 0,

 n>a(ϵ,M )

and lim lim ϵ 2(r −p)/(2−p)

M →∞ ϵ↘0



nr /p−2 P(∥Sn ∥ ≥ ϵ n1/p ) = 0.

n>a(ϵ,M )

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