- Email: [email protected]
Berry–Esseen inequality for linear processes in Hilbert spaces
Statistics & Probability Letters 63 (2003) 243 – 247
Berry–Esseen inequality for linear processes in Hilbert spaces Denis Bosq Universit e Pierre et Marie Curie, Paris, France Received January 2002
Abstract Under mild conditions we obtain the usual Berry–Esseen inequality for linear processes in Hilbert spaces. c 2003 Elsevier Science B.V. All rights reserved. Keywords: Central limit theorem; Linear processes in Hilbert spaces; Berry-Esseen inequality
1. Introduction Let (H; · ) be a separable real Hilbert space and (L; · L ) be the class of bounded linear operators from H to H with its usual uniform norm. Consider a sequence X = (Xn ; n ∈ Z) of H -valued random variables, de3ned on a Probability space (; A; P). X is said to be a (zero-mean) linear process in H (LPH) if Xn =
∞
aj ( n−j );
n ∈ Z;
(1.1)
j=0
where a0 = I (identity), (aj ; j ¿ 1) is a sequence in L and = ( n ; n ∈ Z) is a strong H -white noise (i.e. asequence of i.i.d. H -valued random variables such that 0 ¡ E n 2 ¡ ∞ and E n = 0). ∞ 1 If j=1 aj L ¡ ∞ then the series in (1.1) converges almost surely and in LH (; A; P) (see Denisevskii and Dorogovtsev, 1988 or Bosq, 2000, p. 182). Moreover, X satis3es the central limit theorem (CLT) (cf. Merlev
jaj L ¡ ∞:
j=1
E-mail address: [email protected] (D. Bosq). c 2003 Elsevier Science B.V. All rights reserved. 0167-7152/03/$ - see front matter doi:10.1016/S0167-7152(03)00088-9
(1.2)
244
D. Bosq / Statistics & Probability Letters 63 (2003) 243 – 247
The result is an extension of Theorem 3.1 in Bosq (2000), which provides such an inequality for Hilbertian autoregressive processes of order 1. We state the theorem and discuss its possible extensions in Section 2 while Section 3 is devoted to the proof.
2. The inequality Inthe sequel, ‘∗ denotes adjoint of ‘ ∈ L; C 0 is the autocovariance operator of 0 and A= ∞ j=0 aj . We have the following bound. Theorem 1. Let X be an LPH such that E 0 3 ¡ ∞; AC 0 A∗ = 0 and then n 1 c sup P √ Xi ¡ r − P(N ¡ r) 6 √ ; n ¿ 1; n n r¿0
∞
j=1
jaj L ¡ ∞,
(2.1)
i=1
where N ∼ N(0; AC 0 A∗ ) and c is a positive constant which depends only on the distribution of X. Comments: 1. From the proof (see Section 3) it follows that
c = 6 ’N ∞ E 0
∞
jaj L + c2 ;
j=1
where ’N is density of N and c2 depends only on E 0 3 and CX0 (the covariance operator of X0 ). 2. Condition (1.2) is mild for B.E. bound since ∞ j=1 aj L is sharp for the CLT with normal√ ization n (cf. Merlev
D. Bosq / Statistics & Probability Letters 63 (2003) 243 – 247
245
3. Proof Consider the decomposition n
1 √ Xi = Y n + n ; n i=1 where
Yn = A
and
n
1 √ i n i=1
(3.1)
n n− ‘ aj ( ‘ ) + aj ( ‘ ): n = − ‘=1
j¿n−‘
‘60
j=1−‘
Concerning n we have n n− ‘ En 6 aj L + aj L E 0 ; ‘=1
j¿n−‘
‘ 60
j=1−‘
it follows that ∞ 1 √ E 0 2jaj L ; En 6 n j=1
that is, c1 En 6 √ ; n
n ¿ 1;
(3.2)
where c1 is a 3nite constant. Now, using a BE bound for i.i.d. H -valued random variables (cf. Yurinskii, 1982), one obtains c2 (3.3) sup |P(Yn ¡ r) − P(N ¡ r)| 6 √ ; n ¿ 1; n r¿0 where c2 depends only on E 0 3 and CX0 . Then, consider the quantity n (r) := P(Yn + n ¡ r) − P(Yn ¡ r);
r ¿ 0:
Using elementary inequalities we get |n | 6 P(r − n 6 Yn ¡ r + n ) ∞ 6 P(r − ¡ Yn 6 r + ) dPn ( ): 0
246
D. Bosq / Statistics & Probability Letters 63 (2003) 243 – 247
From (3.3) it follows that r An := P(r − 6 Yn ¡ r + ) dPn ( ) 0
6
r
0
2c2 P(r − 6 N ¡ r + ) dPn ( ) + √ : n
Now, since N has a bounded density, say ’N (cf. Davydov et al., 1998, pp. 97–98), we arrive at the bound 2c2 An 6 2’N ∞ En + √ n and (3.2) yields 2 An 6 √ (’N ∞ c1 + c2 ): n
(3.4)
On the other hand, consider ∞ P(r − 6 Yn ¡ r + ) dPn ( ) Bn := r
=
∞
r
[P(Yn ¡ r) + P(r 6 Yn ¡ r + )] dPn ( ):
We have
c2 Bn 6 P(N ¡ r) + √ P(n ¿ r) n ∞ 2c2 + P(r 6 N ¡ r + ) + √ dPn ( ): n r
Now note that c1 P(N ¡ r)P(n ¿ r) 6 r’N ∞ √ r n and
∞
r
∞ 2c2 2c2 P(r 6 N ¡ r + ) + √ dPn ( ) 6 ’N ∞ dPn ( ) + √ n n r 1 6 √ (’N ∞ c1 + 2c2 ): n
Therefore, 1 Bn 6 √ (c1 ’N ∞ + 3c2 ): n
(3.5)
D. Bosq / Statistics & Probability Letters 63 (2003) 243 – 247
247
So (3.4) and (3.5) entail 1 |n (r)| 6 √ (3c1 ’N ∞ + 5c2 ): n
(3.6)
Finally, (3.1), (3.3) and (3.6) yield n 1 1 Xi ¡ r − P(N ¡ r) 6 √ (3’N ∞ c1 + 6c2 ) sup P √ n i=1 n r¿0 which is (2.1). References Bosq, D., 2000. Linear processes in function spaces. Lecture Notes in Statistics, Vol. 149. Springer, Berlin. Davydov, Y.A., Lifshits, M.A., Smorodina, N.V., 1998. Local Properties of Distributions of Stochastic Functionals, Vol. 173. American Mathematical Society Translations, Providence. Denisevskii, N.A., Dorogovtsev, Y.A., 1988. On the law of large numbers for a linear process in Banach space. Soviet Math. Dokl. 36 (1), 47–50. Merlev
Berry–Esseen inequality for linear processes in Hilbert spaces Denis Bosq Universit e Pierre et Marie Curie, Paris, France Received January 2002
Abstract Under mild conditions we obtain the usual Berry–Esseen inequality for linear processes in Hilbert spaces. c 2003 Elsevier Science B.V. All rights reserved. Keywords: Central limit theorem; Linear processes in Hilbert spaces; Berry-Esseen inequality
1. Introduction Let (H; · ) be a separable real Hilbert space and (L; · L ) be the class of bounded linear operators from H to H with its usual uniform norm. Consider a sequence X = (Xn ; n ∈ Z) of H -valued random variables, de3ned on a Probability space (; A; P). X is said to be a (zero-mean) linear process in H (LPH) if Xn =
∞
aj ( n−j );
n ∈ Z;
(1.1)
j=0
where a0 = I (identity), (aj ; j ¿ 1) is a sequence in L and = ( n ; n ∈ Z) is a strong H -white noise (i.e. asequence of i.i.d. H -valued random variables such that 0 ¡ E n 2 ¡ ∞ and E n = 0). ∞ 1 If j=1 aj L ¡ ∞ then the series in (1.1) converges almost surely and in LH (; A; P) (see Denisevskii and Dorogovtsev, 1988 or Bosq, 2000, p. 182). Moreover, X satis3es the central limit theorem (CLT) (cf. Merlev
jaj L ¡ ∞:
j=1
E-mail address: [email protected] (D. Bosq). c 2003 Elsevier Science B.V. All rights reserved. 0167-7152/03/$ - see front matter doi:10.1016/S0167-7152(03)00088-9
(1.2)
244
D. Bosq / Statistics & Probability Letters 63 (2003) 243 – 247
The result is an extension of Theorem 3.1 in Bosq (2000), which provides such an inequality for Hilbertian autoregressive processes of order 1. We state the theorem and discuss its possible extensions in Section 2 while Section 3 is devoted to the proof.
2. The inequality Inthe sequel, ‘∗ denotes adjoint of ‘ ∈ L; C 0 is the autocovariance operator of 0 and A= ∞ j=0 aj . We have the following bound. Theorem 1. Let X be an LPH such that E 0 3 ¡ ∞; AC 0 A∗ = 0 and then n 1 c sup P √ Xi ¡ r − P(N ¡ r) 6 √ ; n ¿ 1; n n r¿0
∞
j=1
jaj L ¡ ∞,
(2.1)
i=1
where N ∼ N(0; AC 0 A∗ ) and c is a positive constant which depends only on the distribution of X. Comments: 1. From the proof (see Section 3) it follows that
c = 6 ’N ∞ E 0
∞
jaj L + c2 ;
j=1
where ’N is density of N and c2 depends only on E 0 3 and CX0 (the covariance operator of X0 ). 2. Condition (1.2) is mild for B.E. bound since ∞ j=1 aj L is sharp for the CLT with normal√ ization n (cf. Merlev
D. Bosq / Statistics & Probability Letters 63 (2003) 243 – 247
245
3. Proof Consider the decomposition n
1 √ Xi = Y n + n ; n i=1 where
Yn = A
and
n
1 √ i n i=1
(3.1)
n n− ‘ aj ( ‘ ) + aj ( ‘ ): n = − ‘=1
j¿n−‘
‘60
j=1−‘
Concerning n we have n n− ‘ En 6 aj L + aj L E 0 ; ‘=1
j¿n−‘
‘ 60
j=1−‘
it follows that ∞ 1 √ E 0 2jaj L ; En 6 n j=1
that is, c1 En 6 √ ; n
n ¿ 1;
(3.2)
where c1 is a 3nite constant. Now, using a BE bound for i.i.d. H -valued random variables (cf. Yurinskii, 1982), one obtains c2 (3.3) sup |P(Yn ¡ r) − P(N ¡ r)| 6 √ ; n ¿ 1; n r¿0 where c2 depends only on E 0 3 and CX0 . Then, consider the quantity n (r) := P(Yn + n ¡ r) − P(Yn ¡ r);
r ¿ 0:
Using elementary inequalities we get |n | 6 P(r − n 6 Yn ¡ r + n ) ∞ 6 P(r − ¡ Yn 6 r + ) dPn ( ): 0
246
D. Bosq / Statistics & Probability Letters 63 (2003) 243 – 247
From (3.3) it follows that r An := P(r − 6 Yn ¡ r + ) dPn ( ) 0
6
r
0
2c2 P(r − 6 N ¡ r + ) dPn ( ) + √ : n
Now, since N has a bounded density, say ’N (cf. Davydov et al., 1998, pp. 97–98), we arrive at the bound 2c2 An 6 2’N ∞ En + √ n and (3.2) yields 2 An 6 √ (’N ∞ c1 + c2 ): n
(3.4)
On the other hand, consider ∞ P(r − 6 Yn ¡ r + ) dPn ( ) Bn := r
=
∞
r
[P(Yn ¡ r) + P(r 6 Yn ¡ r + )] dPn ( ):
We have
c2 Bn 6 P(N ¡ r) + √ P(n ¿ r) n ∞ 2c2 + P(r 6 N ¡ r + ) + √ dPn ( ): n r
Now note that c1 P(N ¡ r)P(n ¿ r) 6 r’N ∞ √ r n and
∞
r
∞ 2c2 2c2 P(r 6 N ¡ r + ) + √ dPn ( ) 6 ’N ∞ dPn ( ) + √ n n r 1 6 √ (’N ∞ c1 + 2c2 ): n
Therefore, 1 Bn 6 √ (c1 ’N ∞ + 3c2 ): n
(3.5)
D. Bosq / Statistics & Probability Letters 63 (2003) 243 – 247
247
So (3.4) and (3.5) entail 1 |n (r)| 6 √ (3c1 ’N ∞ + 5c2 ): n
(3.6)
Finally, (3.1), (3.3) and (3.6) yield n 1 1 Xi ¡ r − P(N ¡ r) 6 √ (3’N ∞ c1 + 6c2 ) sup P √ n i=1 n r¿0 which is (2.1). References Bosq, D., 2000. Linear processes in function spaces. Lecture Notes in Statistics, Vol. 149. Springer, Berlin. Davydov, Y.A., Lifshits, M.A., Smorodina, N.V., 1998. Local Properties of Distributions of Stochastic Functionals, Vol. 173. American Mathematical Society Translations, Providence. Denisevskii, N.A., Dorogovtsev, Y.A., 1988. On the law of large numbers for a linear process in Banach space. Soviet Math. Dokl. 36 (1), 47–50. Merlev