The aim of this paper is to prove an analogue of Baxter's inequality for
fractional Brownian motion-type processes with Hurst index less than 1/2. This
inequality is concerned with the norm estimate of the difference between
finite- and infinite-past predictor coefficients.Comment: 7 page

Inoue, Akihiko

Kasahara, Yukio

Phartyal, Punam

English

arXiv

We prove an analogue of Baxter's inequality for fractional Brownian motion-type processes with Hurst index less than 1/2. This inequality is concerned with the norm estimate of the difference between finite- and infinite-past predictor coefficients

EPrint Series of Department of Mathematics, Hokkaido University

BAXTER'S INEQUALITY FOR FRACTIONAL BROWNIAN
MOTION-TYPE PROCESSES WITH HURST INDEX
LESS THAN 1=2
AKIHIKO INOUE, YUKIO KASAHARA AND PUNAM PHARTYAL
Abstract. We prove an analogue of Baxter's inequality for fractional Brownian motion-
type processes with Hurst index less than 1=2. This inequality is concerned with the
norm estimate of the di®erence between ¯nite- and in¯nite-past predictor coe±cients.
1. Introduction
To explain Baxter's inequality in the classical setup, we consider a centered, weakly
stationary process (Xk : k 2 Z), and write Áj and Áj;n for the in¯nite- and ¯nite-past
predictor coe±cients, respectively:
P(¡1;¡1]X0 =
X1
j=1
ÁjX¡j ; P[¡n;¡1]X0 =
Xn
j=1
Áj;nX¡j ; (1.1)
where P(¡1;¡1]X0 and P[¡n;¡1]X0 denote the linear least-squares predictors of X0 based on
the observed values fX¡j : j = 1; 2; : : : g and fX¡j : j = 1; : : : ; ng, respectively. There are
many models in which Áj;n's are di±cult to compute exactly while the computation of Áj 's
are relatively easy. In fact, this is usually so for the models with explicit spectral density. It
is known that limn!1 Áj;n = Án (see, e.g., Pourahmadi, 2001, Theorem 7.14). Therefore, it
would be natural to approximate P[¡n;¡1]X0 replacing the ¯nite-past predictor coe±cients
Áj;n by the in¯nite counterparts Áj . Then the error can be estimated by°°°P[¡n;¡1]X0 ¡Xn
j=1
ÁjX¡j
°°° · kX0kXn
j=1
jÁj;n ¡ Áj j; (1.2)
where kZk := E[Z2]1=2. The question thus arises of estimating the right-hand side of (1.2).
Baxter (1962) showed that for short memory processes, there exists a positive constant M
such that Xn
j=1
jÁj;n ¡ Áj j ·M
X1
k=n+1
jÁkj for all n = 1; 2; : : : :
This Baxter's inequality was extended to long memory processes by Inoue and Kasahara
(2006). See also Berk (1974), Cheng and Pourahmadi (1993) and Pourahmadi (2001, Section
7.6.2).
In Inoue and Anh (2007), prediction formulas similar to (1.1) were proved for a class
of continuous-time, centered, stationary-increment, Gaussian processes (X(t) : t 2 R) that
includes fractional Brownian motion (BH(t) : t 2 R) with Hurst index H 2 (0; 1=2) (see
Section 3 for the de¯nition). For
¡1 < t0 · 0 · t1 < t2 <1; t0 < t1; T := t2 ¡ t1; t := t1 ¡ t0; (1.3)
2000 Mathematics Subject Classi¯cation Primary 60G25; Secondary 60G15.
Date: January 17, 2008.
Key words and phrases. Baxter's inequality, fractional Brownian motion, predictor coe±cients.
1
the prediction formulas take the following forms:
P(¡1;t1]X(t2) =
Z 1
0
Ã(s;T )X(t1 ¡ s)ds; P[t0;t1]X(t2) =
Z t
0
Ã(s;T; t)X(t1 ¡ s)ds;
(1.4)
where P(¡1;t1]X(t2) and P[t0;t1]X(t2) are the linear least-squares predictors of X(t2) based
on the in¯nite past fX(s) : ¡1 < s · t1g and ¯nite past fX(s) : t0 · s · t1g, respectively.
The aim of this paper is to prove an analogue of Baxter's inequality for (X(t)). Since
kX(s)k depends on s, a straightforward analogue of (1.2) is not available. Instead, we have°°°°P[t0;t1]X(t2)¡ Z t
0
Ã(s;T )X(t1 ¡ s)ds
°°°° · Z t
0
fÃ(s;T; t)¡ Ã(s;T )gkX(t1 ¡ s)kds:
Here Ã(s;T; t) > Ã(s;T ) > 0 (see Section 3 below). We show that there is a positive
constant M such thatZ t
0
fÃ(s;T; t)¡Ã(s;T )gkX(t1¡ s)kds ·M
Z 1
t
Ã(s;T )kX(t1¡ s)kds for all t ¸ t1; (B)
which we call Baxter's inequality for (X(t)). To the best of our knowledge, this type
of inequality has not been demonstrated before. The key ingredient in the proof is the
representation of the di®erence Ã(s;T; t) ¡ Ã(s;T ) ((3.2) with Proposition 3.2 below). In
fact, we prove a general result that includes (B) (Theorem 4.2 (b)).
2. Fractional Brownian motion
Throughout the paper, we assume 0 < H < 1=2. We can de¯ne the fractional Brownian
motion (BH(t) : t 2 R) with Hurst index H by the moving-average representation
BH(t) =
1
¡(12 +H)
Z 1
¡1
n
((t¡ s)+)H¡ 12 ¡ ((¡s)+)H¡ 12
o
dW (s) (t 2 R);
where (x)+ := max(x; 0) and (W (t) : t 2 R) is the ordinary Brownian motion. In this
section, we study the di®erence between the ¯nite- and in¯nite-past predictor coe±cients
of (BH(t)).
Let t0; t1; t2; t and T be as in (1.3). We de¯ne the in¯nite- and ¯nite-past predictors
P(¡1;t1]BH(t2) and P[t0;t1]BH(t2) of (BH(t)), respectively, as we de¯ned in Section 1 for
(X(t)). The following prediction formulas, that is, (1.4) for (BH(t)), were established by
Yaglom (1955) and Nuzman and Poor (2000, Theorem 4.4), respectively (see also Anh and
Inoue, 2004, Theorem 1):
P(¡1;t1]BH(t2) =
Z 1
0
Ã0(s;T )BH(t1¡s)ds; P[t0;t1]BH(t2) =
Z t
0
Ã0(s;T; t)BH(t1¡s)ds;
where
Ã0(s;T ) =
cos(¼H)
¼
1
s+ T
µ
T
s
¶ 1
2+H
(0 < s <1);
Ã0(s;T; t) =
cos(¼H)
¼
"
1
s+ T
µ
T
s
¶ 1
2+H
µ
t¡ s
t+ T
¶ 1
2¡H
+ ( 12 ¡H)B tt+T (H + 12 ; 1¡ 2H)
1
t
½µ
t
s
¶µ
t
t¡ s
¶¾ 1
2+H
#
(0 < s < t);
with Bs(p; q) :=
R s
0
up¡1(1¡ u)q¡1du being the incomplete beta function.
2
Throughout the paper, f(t) » g(t) as t ! 1 means limt!1 f(t)=g(t) = 1. A positive
measurable function f , de¯ned on some neighbourhood [M;1) of 1, is called regularly
varying with index ½ 2 R, written f 2 R½, if for all ¸ 2 (0;1), limt!1 f(t¸)=f(t) = ¸½.
When ½ = 0, we say that the function is slowly varying. A generic slowly varying function
is usually denoted by `. See Bingham et al. (1989) for details. The function kBH(t1 ¡ s)k
of s is in RH since kBH(s)k = jsjHkBH(1)k for s 2 R.
We will use the next lemma in Section 4. For 0 < H < 12 and ½ > ¡ 12 +H, we put
C(H; ½) := 1¡ ½B( 12 ¡H + ½; 12 ¡H)
1¡ 2H
1 + 2H
;
where B(p; q) :=
R 1
0
up¡1(1¡ u)q¡1du denotes the beta function.
Lemma 2.1. Let g be locally bounded in [0;1) and g 2 R½ with ½ > ¡ 12 +H. Then, for
¯xed T > 0,Z t
0
fÃ0(s;T; t)¡ Ã0(s;T )gg(s)ds » C(H; ½)1
2 +H ¡ ½
¢ t Ã0(t;T )g(t) (t!1):
Proof. If t is large enough, then g(t) > 0. For such t, we have, by simple computation,
1
t Ã0(t;T )g(t)
Z t
0
fÃ0(s;T; t)¡ Ã0(s;T )gg(s)ds =
Z 1
0
Ã0(ts;T; t)¡ Ã0(ts;T )
Ã0(t; T )
g(ts)
g(t)
ds
=
Z 1
0
I(s;T; t)
g(ts)
g(t)
ds+
Z 1
0
II(s;T; t)
g(ts)
g(t)
ds;
where
I(s;T; t) = s¡
1
2¡H 1 + (T=t)
s+ (T=t)
"µ
1¡ s
1 + (T=t)
¶ 1
2¡H
¡ 1
#
;
II(s;T; t) = ( 12 ¡H)B tt+T (H + 12 ; 1¡ 2H)(t=T )
1
2+Hf1 + (T=t)g fs(1¡ s)g¡ 12¡H :
Since Bt=(t+T )(H + 12 ; 1 ¡ 2H) »
¡
1
2 +H
¢¡1 (T=t) 12+H as t ! 1, we easily see that, for
0 < s < 1,
jI(s;T; t)j · const.£s¡ 12¡H ; jII(s;T; t)j · const.£fs(1¡s)g¡ 12¡H (t large enough):
Put ± = 12 (
1
2 ¡H + ½) > 0. Then, for 0 < s < 1, also we have
jg(ts)=g(t)j · 2s½¡± (t large enough) (2.1)
(cf. Bingham et al., 1989, Theorem 1.5.2). Therefore, the dominated convergence theorem
yields, as t!1,Z 1
0
I(s;T; t)
g(ts)
g(t)
ds!
Z 1
0
(1¡ s) 12¡H ¡ 1
s
3
2+H¡½
ds =
1¡ ( 12 ¡H)B( 12 ¡H + ½; 12 ¡H)
1
2 +H ¡ ½
; (2.2)Z 1
0
II(s;T; t)
g(ts)
g(t)
ds! (
1
2 ¡H)B( 12 ¡H + ½; 12 ¡H)
1
2 +H
: (2.3)
In (2.2), we have used integration by parts. From (2.2) and (2.3), we obtain the lemma. ¤
Remark 2.2. From Lemma 2.1 with g(t) = kBH(t1 ¡ t)k, whence ½ = H, we see thatZ t
0
fÃ0(s;T; t)¡ Ã0(s;T )gkBH(t1 ¡ s)kds » 2
¼
cos(¼H)C(H;H)T
1
2+HkBH(1)k ¢ t¡ 12
(t!1):
3
It is interesting that the order of decay here is t¡1=2, whence does not depend on H.
3. Fractional Brownian motion-type processes
In this and next sections, we consider the predictor coe±cients for the fractional Brow-
nian motion-type process (X(t) : t 2 R) in Inoue and Anh (2007). It is a stationary-
increment Gaussian process de¯ned by
X(t) =
Z 1
¡1
fc(t¡ s)¡ c(¡s)g dW (s); (t 2 R);
where the moving-average coe±cient c is a function of the form
c(t) =
Z 1
0
e¡tsº(ds) (t > 0); = 0 (t · 0)
with º being a Borel measure on (0;1) satisfying R1
0
(1+ s)¡1º(ds) <1. We also assume
lim
t!0+
c(t) =1; c(t) = O(tq) (t! 0+) for some q > ¡1=2,
c(t) » 1
¡(12 +H)
t¡(
1
2¡H)`(t) (t!1);
where `(¢) is a slowly varying function and H 2 (0; 1=2).
The process (X(t)) also has the autoregressive coe±cient a de¯ned by a(t) := ¡(d®=dt)(t)
for t > 0, where ® is the unique function on (0;1) satisfying
¡iz
µZ 1
0
eiztc(t)dt
¶µZ 1
0
eizt®(t)dt
¶
= 1 (=z > 0):
We know that a(t) =
R1
0
e¡tss¹(ds) for some Borel measure ¹ on (0;1) (see Inoue and
Anh, 2007, Corollary 3.3). In particular, a is also positive and decreasing on (0;1). By
Inoue and Anh (2007, (3.12)), we have
a(t) » t
¡( 32+H)
`(t)
¢ (
1
2 +H)
¡( 12 ¡H)
(t!1): (3.1)
Example 3.1. If º is given by º(ds) = ¼¡1 cos(¼H)s¡(
1
2+H)ds on (0;1), then c(t) =
t¡(
1
2¡H)=¡( 12 + H) for t > 0, whence (X(t)) reduces to (BH(t)). In this case, a(t) =
t¡(
3
2+H)( 12 +H)=¡(
1
2 ¡H).
We refer to Inoue and Anh (2007, Example 2.6) for another example of (X(t)) which
has two di®erent indexes H0 and H describing its path properties and long-time behaviour,
respectively.
We put
b(s; u) :=
Z u
0
c(u¡ v)a(s+ v)dv (s; u > 0):
For k = 1; 2 : : : and s; t; T > 0, we de¯ne bk(s; t;T ) iteratively by
b1(s;T; t) := b(s; T ); bk(s;T; t) :=
Z 1
0
b(s; u)bk¡1(t+ u;T; t)du (k = 2; 3; : : : ):
Note that bk's are positive because both c and a are so. By Inoue and Anh (2007, Theorems
3.7 and 1.1), the in¯nite- and ¯nite-past predictor coe±cients Ã(s;T ) and Ã(s;T; t) in (1.4)
4
are given, respectively, by
Ã(s;T ) = b(s; T ) = b1(s;T; t) (s > 0);
Ã(s;T; t) =
1X
k=1
fb2k¡1(s;T; t) + b2k(t¡ s;T; t)g (0 < s < t):
Notice that Ã(s;T; t) here corresponds to h(t¡ s;T; t) in Inoue and Anh (2007). We have
Ã(s;T; t)¡ Ã(s;T ) =
1X
k=1
fb2k(t¡ s;T; t) + b2k+1(s;T; t)g (0 < s < t); (3.2)
which plays a key role in the proof of Baxter's inequality (B) in the next section.
To prove Baxter's inequality (B), we need to discuss the following. Consider
¯(t) :=
Z 1
0
c(v)a(t+ v)dv (t > 0);
and de¯ne ±k(t; u; v) for k = 1; 2; 3; : : : and t; u; v > 0, iteratively by
±1(t; u; v) := ¯(t+u+v); ±k(t; u; v) :=
Z 1
0
¯(t+v+w)±k¡1(t; u; w)dw (k = 2; 3; : : :):
Proposition 3.2. For s; t; T > 0 and k ¸ 2,
bk(s;T; t) =
Z T
0
c(T ¡ v)dv
Z 1
0
a(s+ u)±k¡1(t; u; v)du:
This can be proved in the same as in Inoue and Kasahara (2006, Theorem 2.8); we omit
the proof.
Next, we give some results on the asymptotic behaviour of ±k's. For k = 1; 2; : : : and
u ¸ 0, we de¯ne fk(u) iteratively by
f1(u) :=
1
¼(1 + u)
; fk(u) :=
Z 1
0
fk¡1(u+ v)
¼(1 + v)
dv (k = 2; 3; : : :):
Proposition 3.3. (a) For r 2 (1;1), there exists N > 0 such that 0 < ±k(t; u; v) ·
fk(0)fr cos(¼H)gkt¡1 for u; v > 0; k 2 N; t ¸ N .
(b) For k 2 N and u; v > 0, ±k(t; tu; v) » t¡1fk(u) cosk(¼H) as t!1.
This can be proved in the same as in Inoue and Kasahara (2006, Proposition 3.2); we
omit the proof.
4. Baxter's inequality
In this section, we prove Baxter's inequality (B). Let (X(t)), Ã(s;T ) and Ã(s;T; t) be as
in Section 3. Since a is decreasing, we have a(T + t)
R T
0
c(v)dv · Ã(t;T ) · a(t) R T
0
c(v)dv,
so that (3.1) implies
Ã(t;T ) » a(t)
Z T
0
c(v)dv » t
¡( 32+H)
`(t)
¢ (
1
2 +H)
¡(12 ¡H)
Z T
0
c(v)dv (t!1): (4.1)
Here is the extension of Lemma 2.1 to (X(t)).
Lemma 4.1. Lemma 2.1 with Ã0(s;T; t) and Ã0(s;T ) replaced by Ã(s;T; t) and Ã(s;T ),
respectively, holds.
5
Proof. For t large enough, using (3.2), we may write
D(t) :=
1
t Ã(t;T )g(t)
Z t
0
fÃ(s;T; t)¡ Ã(s;T )gg(s)ds =
Z 1
0
Ã(ts;T; t)¡ Ã(ts;T )
Ã(t;T )
g(ts)
g(t)
ds
=
1X
k=1
Z 1
0
b2k(ts;T; t)
Ã(t;T )
g(t(1¡ s))
g(t)
ds+
1X
k=1
Z 1
0
b2k+1(ts;T; t)
Ã(t;T )
g(ts)
g(t)
ds
and
bk(ts;T; t)
Ã(t;T )
=
a(t)
Ã(t;T )
Z T
0
c(T ¡ v)dv
Z 1
0
a(ts+ u)
a(t)
±k¡1(t; u; v)du
=
a(t)
Ã(t;T )
Z T
0
c(T ¡ v)dv
Z 1
0
a(t(s+ u))
a(t)
¢ t ±k(t; tu; v)du:
Put ± = 13 minf 12 ¡H; 12 ¡H + ½g > 0. By (3.1), we have a 2 R¡(3=2)¡H , and
a(t¸)=a(t) · 2¸¡ 32¡H¡± for 0 < ¸ < 1; · 2¸¡ 32 for ¸ > 1 (t large enough)
(cf. Bingham et al., 1989, Theorems 1.5.2 and 1.5.6). Choose 0 < r < 1= cos(¼H) so that
x := r cos(¼k) 2 (0; 1). Then, by Proposition 3.3 (a), we have for 0 < s < 1 and v > 0,Z 1
0
a(t(s+ u))
a(t)
¢ t ±k(t; tu; v)du · 2fk(0)xk
·Z 1¡s
0
du
(s+ u)
3
2+H+±
+
Z 1
1¡s
du
(s+ u)¡
3
2
¸
· 2fk(0)xk
"
s¡
1
2¡H¡±
1
2 +H + ±
+ 2
#
(t large enough):
By Inoue and Kasahara (2006, Lemma 3.1),
P1
k=0 fk(0)x
k < 1. From these facts as well
as (2.1), (4.1), Proposition 3.3 (b) and the dominated convergence theorem, we see that
limt!1D(t) = D, where
D :=
1X
k=1
cos2k¡1(¼H)
Z 1
0
½Z 1
0
f2k¡1(u)
(s+ u)
3
2+H
du
¾
(1¡ s)½ds
+
1X
k=1
cos2k(¼H)
Z 1
0
½Z 1
0
f2k(u)
(s+ u)
3
2+H
du
¾
s½ds:
Since (BH(t)) is a special case of (X(t)), this also holds for Ã0(t;T ) and Ã0(s;T; t). There-
fore, from Lemma 2.1, we conclude that D = C(H; ½)=( 12 + H ¡ ½). Thus the lemma
follows. ¤
Following theorems are the conclusion of this paper.
Theorem 4.2. Let g be locally bounded in [0;1) and g 2 R½ with ½ 2 (¡ 12 +H; 12 +H).
(a) For ¯xed T > 0, we haveZ t
0
fÃ(s; T ; t)¡ Ã(s;T )gg(s)ds » C(H; ½)
Z 1
t
Ã(s;T )g(s)ds (t!1):
(b) There exists a positive constant M such thatZ t
0
fÃ(s; T ; t)¡ Ã(s;T )gg(s)ds ·M
Z 1
t
Ã(s;T )g(s)ds (t > 1):
6
Proof. By (4.1), the function Ã(s;T )g(s) in s belongs to R½¡ 32¡H . Since ½ <
1
2 + H, we
have Z 1
t
Ã(s;T )g(s)ds » 11
2 +H ¡ ½
tÃ(t;T )g(t) (t!1):
The assertion (a) follows from this and Lemma 4.1, while (b) from (a). ¤
Theorem 4.3. (a) Baxter's inequality (B) holds.
(b) For ¯xed T > 0, we have, as t!1,Z t
0
fÃ(s; T ; t)¡ Ã(s;T )gkX(t1 ¡ s)kds » C(H;H) 1 + 2H¡( 12 ¡H)
ÃZ T
0
c(v)dv
!
kBH(1)k ¢ t¡ 12 :
Proof. By Inoue and Anh (2007, Lemma 2.7), kX(t)k » kBH(1)k tH`(t) as t!1. So, (a)
follows from Theorem 4.2 (b) if we put g(s) := kX(t1¡s)k = kX(s¡ t1)k. Also, (b) follows
from Lemma 4.1 and (4.1). ¤
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formation. J. Appl. Probab. 37, 429{452.
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Interscience, New York.
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(2) 8 (1958), 87{141.
E-mail address: inoue@math.sci.hokudai.ac.jp
E-mail address: y-kasa@math.sci.hokudai.ac.jp
E-mail address: punamphartyal@gmail.com
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810,
Japan
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Baxter's inequality for fractional Brownian motion-type processes with Hurst index less than 1/2

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Baxter's inequality for fractional Brownian motion-type processes with Hurst index less than 1/2

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