- Email: [email protected]
A Hanson-Russo-type law of the iterated logarithm for fractional Brownian motion
Statistics & Probability North-Holland
Letters
17 (1993) 27-34
3 May 1993
A Hanson-Russo-type law of the iterated logarithm for fractional Brownian motion Charles
El-Nouty
L.S. T.A. Universite’ Paris VI, Paris, France Received March 1992 Revised September 1992
Abstract: Let B,(t) be a fractional Brownian motion with index 0 < H < 1. We investigate the set _Y of almost sure limit points of the sequence of functions &(Bn(n + tg(n))BH(n)), where g(n) and p, are suitably chosen functions of n and 0 Q t G 1. Our results give a version of the Hanson-Russo law of the iterated logarithm for general fractional Brownian motions. AMS 1980 Subject Classifications:
Keywords:
fractional
Brownian
60F15, 60GlS.
motion;
law of the iterated
logarithm;
reproducing
kernel
Hilbert
space.
1. Introduction
Let {w(t), 0 G t < w) be a standard Wiener process. Many authors have studied the behavior of increments of the form W(T) - W(T - ar) where 0 G uT G T is a function of T > 0 satisfying suitable regularity and growth conditions. Csorg6 and R&&z (19791, for instance, have given conditions on aT which imply that lim sup T-+~ a~c?r!-+
IW(t+a,) (2a,[ln
-w(t)1
T/a,+
In, T])“’
whereas Book and Shore (1978) have characterized W(t+a,)
lim inf
sup =+CC O
(2u,[ln
= 1
the cases where, for 0 < r < 03, l/2
-W(t)1
T/u,
(1.1)
ase
+ In, T])1’2
= t--i r:
1
a’s*
(1.2)
Here we set In u = log(u V e) and In, u = l&n u) for u > 0. Chen, Kong and Lin (1986) and Hanson and Russo (1983a, 1983b, 1989) have followed the same lines of investigation by considering statistics of the form given in (1.1)~(1.21, with the supremum over 10, T - aTI replaced by the supremum over smaller families of sub-intervals of 10, Tl. Furthermore, assuming suitable growth and regularity conditions imposed on uT, they proved that results of the type
Correspondence
to: Charles
0167-7152/93/$06.00
El-Nouty,
185 bvd Vincent
0 1993 - Elsevier
Science
Auriol,
Publishers
75013 Paris, France.
B.V. All rights reserved
27
Volume
17. Number
STATISTICS&PROBABILITY
1
3 May 1993
LETTERS
given in (l.l)-(1.2) were also valid for other functionals than that considered in these statements. We refer to Hanson and Russo (1989) and the references therein for further details and limit ourselves to the following result, commonly referred to as the Hanson-Russo law of the iterated logarithm. We have almost surely W(T) - W(T-t) lim inf T-CC
E [I,
sup +
(2t[ln T/t + In, t])12
w41
i =F(a)
whenever lim inf q/T T+-
whenever q-/T + a,
= a, (1.3)
where F(u)
= i
0
if a =O,
-1/{1-f(loga)}1’2
ifO
Let (B,(t), 0 < t < 1) be a fractional Brownian motion with index 0 < H < 1, i.e. a centered Gaussian process with stationary increments such that B,(O) = 0 a.s. and satisfying E(II,(c))~ = t2H (see e.g. Taqqu, 1977; Ortega, 1989; and the references therein for further details on the properties of BH). For H = i, {BH(t), 0 d t 6 1) so defined is a standard Wiener process. Denote by K, the unit ball of the reproducing kernel Hilbert space II pertaining to BH( *) and endowed with the norm II IIr. Taqqu and Czado (1985) have considered these spaces in the general framework of self-similar processes. Let C[O, 11 denote the set of continuous functions on [O, 11, endowed with the topology of uniform convergence. Whenever a sequence {f,, IZ> 1) of functions in C[O, 11 is relatively compact, we denote by _P(f,) the limit set of {f,}, i.e. the set of all limits of convergent subsequences Cf,,} of f,, The celebrated Strassen (1964) functional law of the iterated logarithm (LIL) shows that, if f,(t) = (2n In, n)-‘/2W(nt), then the sequence {f,, II z 1) is almost surely relatively compact with limit set .-Y(f,J = K1,2. Taqqu (1977) established a version of Strassen’s functional LIL for fractional Brownian motion. Namely he showed that BH(nt)
_!z
=K,
(1.4)
a.e.
nH(2 In, n)1’2 1
R&&z (1979) obtained a functional form of Strassen’s LIL, appropriate to (1.1). The main purpose of this paper is to prove a functional version of the Hanson-Russo LIL (1.3), in the spirit of R&&z (1979) for the fractional Brownian motion BH. Our main results are given in the following Section 2. The proofs of our theorems are postponed to Section 3.
2. Results To motivate further our statements, we first outline the arguments used by Taqqu (1977) to prove the LIL for BH. Towards the aim of proving (1.4), Taqqu (1977) showed in the first place by general methods that _Y(f,J c K, a.e. with f,(t) = BH(nt)/{nH(2 In n)i/‘}. Then, he achieved equality of these sets by showing that _Y(f,*) = K, a.e. for a suitably chosen subsequence. We will follow roughly the same arguments, with a different method for the inclusion -!Z’(f,) c KH. Towards proving this statement, we will use the results of Lai (1973, 1974) stated in Theorem A below: Theorem A. Let {X(t); 0 G t G l} be a separable centered Gaussian process with continuous covuriance function R(s, t 1 satisfying JW(t) 28
-x(4)2
Ogt
O
(2.1)
Volume 17, Number 1
STATISTICS&PROBABILITY
LETTERS
3 May 1993
where W is a continuous nondecreasing function on [O, 11 such that m
/1
‘Z’(e-‘)
du < +m.
Let {X,,(t >, n 2 l} be a sequence of Gaussian processes defined on the same probability space and having the same distribution as the process X(t), and let Y,(t)
= (2 In n)-1’2Xn(
t).
Then, with probability one, the sequence {Y,(t), n > 3} is relatively compact in C[O, 11 with set of limit points contained in the unit ball K of the reproducing kernel Hilbert space pertaining to R. Letting F,={Xi(t):Ogt~l,l~j
and
F=
fi F,, n=l
suppose furthermore
that F is a Gaussian family of random variables such that
n+m~~n_r,E(E(Xm(t)
I F,))2 = 0
for each t E [0, 11.
Then with probability one, the set of limit points of {Y,(t),
n 2 3) in C[O, 11 IYS equal to the set K.
Remark. It is readily verified that B,(t) follows the condition (2.1), since E(B,(t) Let g(u) be a positive nondecreasing function on [0, 03[. Set, for n 2 1,
X(f) =
BH(n
+
@b>) -BH(n)
swH
)
(2.2)
- B,(s))~
q
= ) t -s I 2H.
o
Our main results are stated in the following two theorems. Theorem 1. (i) Zf g(u) = o(ue) as u + 03 for all fuced p > 0, then, with probability one, the sequence ((2 In n)-1/2X,(t), n 2 1) is relatively compact in C[O, 11 and its set of limit points in C[O, l] is equal to K “iii) Zf g(u) = ua$( u ) as u -+ a for some 0 <(Y < 1, +(u) + (I)(U))-’ = o(ua> as u + 03 for all fixed p > 0 and if lim,, Ilim,_,,p”~(pu>/Jl(u> = 1, then, with probability one, the sequence {(2(1(Y>In n)-‘/2X n(t) 7 n > l] is relatively compact in C[O, l] with set of limit points in C[O, l] equal to K,. Theorem 2. (i) sequence {(2 In, (ii) Zf g(u) = sequence {(2(1 + contained in K,.
Zf g(u) = ~(1 + o(1)) as 1.4+ 03 for some 0-1’2X,(t>, n 3 l] k relatively compact in C[O, 11 with set of limit points contained in K,. At&n ujWp(l + o(l)) as u -+ 03 for some A > 0 and /3 > 0, then, with probability one, the p> In, n)- ‘12Xn(t), n >, l} is relatively compact in C[O, I] with set of limit points in C[O, l]
3. Proofs c K, a.e. and the sequence ((2 In n)-‘/‘X,(*)} is a.s. relatively compact in C[O, 11. To prove the reverse inclusion, we will use Theorem 4.1. of Mangano (1976) for a well-chosen subsequence. This result shows that Theorem Proof of Theorem 1. To prove (i), we first note that, by Theorem A, _Y((2 In n)-‘/2X,(*))
29
Volume
17, Number
1
STATISTICS
&PROBABILITY
LETTERS
3 May 1993
A remains true when condition (2.2) is replaced by a weaker condition of asymptotic independence. have here
vw)xnw)
=
2g(m)ig(n)H [(m-n+fg(m))2H+
We
lm-n-sg(n)12H
For m 2 n and g(m) = o(m - n) as n --f m, a second order expansion yields E(X,(t)X,(s))
-H(2H-
1)ts
M4s(411-” [(m -n)2]1-H
’
(3-l)
Choose any p > 1, and set nk = [ kJ’] for k = 1, 2.. . . Since g(u) = o(zL(~-~)/~) and nk+l - nk -pkpP1, we have g(n,+,) = o(nk+l - nk> as k + w. We will now show that for each k E N*, there exists an integer k, a k such that 2[kP]g[(k+r)P]
forall
rzk,-k.
(3.2)
Since r > k, -k implies k,P - 1 G [k,P] G [(k + rjP], we see that (3.2) holds whenever 2kP f k,P - 1, or equivalently k, > (2kP + l)‘lp. Next, we prove that for any integer k’ > k + r, we have nk’
-n,ar.
(3.3)
To prove (3.3), we start with the obvious inequalities (following from an application of the mean-value theorem to the function x +xp on [k, k’l) pkp-‘(
k’ - k) < ktp - kP
k’ - k).
Since k ” - kP G [k’P] - [kP] + 1 = nkt - nk + 1, we see from this inequality combined with k’ - k > r that (3.3) holds whenever k’ - k
k’ - k) - 1~ nkt - nk.
Finally, since k’ - k z 1, the latter inequalities are satisfied for k a (2/p)‘/(p-1). By (3.1), we see that for any E > 0, there exists an integer k, such that for each integer k a k, and for each integer k’ > k + r with r z k, - k,
(3.4) Observe that dnk’)dnk) (nkf
30
-nk)2
=p dnkf)
gbk)
nk’-nk
(nkr-nk)1’2
1 (nkt-nk)1’2’
STATISTICS&PROBABILITY LETTERS
Volume 17, Number 1
3 May 1993
nk). We deduce from (3.2) that 2n, G nkl and Since nkl -nk’-l & nk’ -nk we have g(n,,)=o(n,,consequently ny’ < (n,, - nk)l12. Since g(n,) = o,we obtain g(n,) = o((n,, - r~,)~/‘). Thus, we have
g(n&(nk)
~
1 (3.5)
cnk’
-nk)2
(nk’-nk)1’2’
By combining (3.31, (3.4) and (3.51, we see that for any E > 0, there exists an integer k, such that for each integer k > k, and for each integer k’ > k + r with r 2 k, - k,
_‘, k
and therefore
It readily follows that
Hence, by Theorem 4.1 of Mangano (1976), P((2 In k)-1/2Xn,(. k --) 03,we obtain that _~?((2 In n)-l/*X,(
e)) IP-~‘~K,
1) = K,
Lemma.
nk
N
P In k as
a.e.
But p > 1 is arbitrary, and so _Y((2 In n)- ‘12Xn( *)) = K, a.e. To prove (ii), we consider 1 < 6 < l/(1 - a) and mk = [k’l.
P((2 In k)-1/2X m!i(.)I cK
a.e. Since In
BY
Lai
(1%‘4),
we
have
H’
We have uniformly over mk < m < mk+ ,,
+
SUP O
2g(mk)-H
i&(m+&(m))
-BH(mk+tg(mk))i’
proof. Let t be fixed. By combining the triangular inequality and the fact that g is nondecreasing, follows that BH(m
+
@trn))
-hdrn)
BH(mk
+
gwH BH(mk
-BH(mk)
dmk)H +
@tmk))
-BH(mk)
<
-
g(m)” BH(m
&tmk))
it
+
@cm))
+ swH
Bh(mk
ffg(mk)) dmk)
-BH(m)
-
-‘fdmk) H
BH(mk+tg(mk))-BH(mk) g(m)”
31
Volume
STATISTICS&PROBABILITY
17, Number 1
l-
+
g(m,) ~ ( g(m)
H 11
BH( m + fg( m)) -BH(mk
+ ‘dmk>)
II
@klH
This proves the lemma.
3 May 1993
LETTERS
+
BH(m)
-BH(mk) dmkjH
’
0 k a 3) is relatively compact in
Proof of Theorem 1 (continued). Since the sequence ((2 In k)-‘/2X,k(*), C[O, 11 with probability one, it follows that sup II(2 In k) -“2xm,(
0) II<
m
a.e.
k>3
Moreover because g is nondecreasing
and since
we have
lim
(3.6)
k-m
mk + dmk+l) - dmk). Since mk+l - mk N 6ksm1 and g(m,) Let Pkcmk+lpk = o(g(mk)). We now need the following bounds.
P
sup
- k**$([k*]),
we have
h k)1’2 IBH(m+r(g(m))-BH( mk+tg(mk))l>,i?g(mk)H(2
mk
O
J
r
I&(t)-&(S)1 sup m,-g(m,+,)ds,t~mk+l-g(m&) lS--fl
=P
sup
1BH(t)
-hdS)
>cg(mk)H(21n
1 2
(Pk)
O~s,tgm&+,--m&+2g(mk+,)
HE
% [
is-ti
k)1’2
1 H(2
In
By Lemma 5 of Ortega (1989), the exponential term of the majoration is therefore
k)l12
I.
For E > 0, we can choose k so large that the above exponent is greater than one. Thus, we have lim k-tm
sup mk
O&t<1
32
g(mk)-H(2
In k)-1’2
1 BH(m
+ 6drn>)
-BH(
mk + tg(m,))
I= 0
a.e.
Volume 17, Number 1
STATISTICS&PROBABILITY
LETTERS
3 May 1993
Thus, by combining the lemma and (3.6), we obtain that lim
sup
k-too mk
(2 In k)-l’*
Since _Y((2 In k)-‘/*X,k(.))
c K,,
11X,( *) -Xmk( *) II= 0
we have c~-~/*K,
_5?((2 In n)-l/*X,,(.))
a.e.
a.e.
for any 1
Letting S 7 (1 - (~)-l, we so obtain that _Y((Z In n) -“*XJ
*)) c (1 - a)l’*KH.
To prove the reverse inclusion, we use a similar argument as that given for (i), making use of the fact that g(m) = o(m -n). We also choose the subsequence nk = [kPl for k & 1 such that p > (1 - a)-l. 0 Proof of Theorem 2. To prove (i), for any fixed E > 0, we choose c > 1 such that E*CZ*~> 4(2(c - 1X1 + CZ>>*~. Let also mk = [c kI. For mk (c - 1 + 2ac)/(c - 1X1 + a). It follows from these assumptions that for all large k, P [
IBH(m+tg(m))--BH(m,+tg(mk))I sup l?Zk
>g(mk)Ha(21n
k)1’2
Ogta1
IBH(t) -BH(S)
sup 0~s,t~mk+,-mk+2g(m,+l Is-tlGPk [
&
=qP
h
k)l’* I
I BH(f)
sup v=l
1 ag(mk)Hs(2
)
-B&s)
I ag(mk)H@
h
k)“*
I b-l)Pk4srt<(v+l)p~ SUP
IBH(t)
-B,(S))
k)l’*
ag(mk)Hc(2h
[ Oe,t<2pk
I BH(t)
I ag(m,)“t(2
In k)“2
1.
1
1
By Ortega (1989), the exponential term of the majoration is therefore
But a
2(c-1)(1+(Y)
1 2H
0
‘l
~*~x*~>4(2(c-l)(l+a))*~.
Thus we have
lim sup k-m
SUP
[ g(mk)H(2
h
k)l’*lpl
I BH(m
+
e(m))
-BH(mk
+
@drnk>)
I GE.
m~
O
33
Volume
17, Number
STATISTICS&PROBABILITY
1
LETTERS
3 May 1993
a.e. By all this and the lemma, we obtain that lim sup k+-
sup m~
“x~(*)-xmx(.)” (2 In k)“2
< (‘“-l)
limsup(2In k+m
CH
k)-1’2((X
“k
(*)((+E
o
1)
(P-
G
SE. CH
By Theorem A, we have _‘5?((2In k)-‘/2Xm,!*)) cK, a.e. Since ln k N h2mk as k + ~0, we have _!Z((2 ln2m,)-‘/2Xmk(~)) c K,. Hence, since E > 0 can be chosen arbitrary small, we have
To prove (ii), take 6 = (1 -i-p)-‘. We choose c > 0 such that &2A2H> 4(2~a+‘6)~~ and mk = [exp(ck’)]. pk = mk+ 1 - mk + dmk+ 1) - dm,>, we have pk - ci3k-‘P exp(ck*). The proof follows along the same lines as the proof of (9. q htting
Acknowledgment
I am grateful to Professor P. Deheuvels for his helpful comments and also to the referee remarks that have led to improvements in this work.
for several
References Book, S.A. and T.R. Shore (19781, On large intervals in the CsiirgB-R&&z theorem on increments of a Wiener process, Z. Wuhrsch. Verrv. Gebiete 46, l-11. Chen G., F. Kong and Z. Lin (19861, Answers to some questions about increments of a Wiener process, Ann. Probab. 14, 1252-1261. CsiirgB M. and P. Rev&z (1979), How big are the increments of a Wiener process? Ann. Probab. 7, 731-737. Hanson D.L. and R.P. Russo (1983a), Some results on increments of the Wiener process with applications to lag sums of i.i.d. random variables, Ann. Probab. 11, 609-623. Hanson, D.L. and R.P. Russo (1983b), Some more results on increments of the Wiener process, Ann. Probab. 11, 10091015. Hanson D.L. and R.P. Russo (1989). Some “lim inf’ results for increments of a Wiener process, Ann. Probub. 17, 1063-1082. Lai T.L. (19731, On Strassen-type laws of the iterated logarithm for delayed averages of the Wiener process, Bull. Inst. Math. Acad. Sinica 1, 29-39.
34
Lai T.L. (19741, Reproducing Kernel Hilbert spaces and the LIL for Gaussian processes, Z. Wahrsch. Verw. Gebiete 29, 7-19. Mangano G.C. (19761, On Strassen-type laws of the iterated logarithm for Gaussian elements in abstract spaces, Z. Wuhrsch. Verw. Gebiete 36, 227-239. Ortega J. (1989), Upper classes for the increments of fractional Brownian motion, Probab. Theory. Rel. Fields 80, 365-379. Revesz P. (19791, A generalization of Strassen’s functional law of iterated logarithm, Z. Wahrsch. Verw. Gebiete 50, 257264. Strassen V. (1964), An invariance principle for the law of the iterated logarithm, Z. Wuhrsch. Verw. Gebiete 3, 211-226. Taqqu M.S. (1977), LIL for sums of non-linear functions of Gaussian variables that exhibit a long-range dependence, Z. Wuhrsch. Verw. Gebiete 40, 203-238. Taqqu M.S. and C. Czado (19851, A survey of functional laws of the iterated logarithm for self-similar processes, Comm. Statist.-Stochastic Models. 1, 77-115.
Letters
17 (1993) 27-34
3 May 1993
A Hanson-Russo-type law of the iterated logarithm for fractional Brownian motion Charles
El-Nouty
L.S. T.A. Universite’ Paris VI, Paris, France Received March 1992 Revised September 1992
Abstract: Let B,(t) be a fractional Brownian motion with index 0 < H < 1. We investigate the set _Y of almost sure limit points of the sequence of functions &(Bn(n + tg(n))BH(n)), where g(n) and p, are suitably chosen functions of n and 0 Q t G 1. Our results give a version of the Hanson-Russo law of the iterated logarithm for general fractional Brownian motions. AMS 1980 Subject Classifications:
Keywords:
fractional
Brownian
60F15, 60GlS.
motion;
law of the iterated
logarithm;
reproducing
kernel
Hilbert
space.
1. Introduction
Let {w(t), 0 G t < w) be a standard Wiener process. Many authors have studied the behavior of increments of the form W(T) - W(T - ar) where 0 G uT G T is a function of T > 0 satisfying suitable regularity and growth conditions. Csorg6 and R&&z (19791, for instance, have given conditions on aT which imply that lim sup T-+~ a~c?r!-+
IW(t+a,) (2a,[ln
-w(t)1
T/a,+
In, T])“’
whereas Book and Shore (1978) have characterized W(t+a,)
lim inf
sup =+CC O
(2u,[ln
= 1
the cases where, for 0 < r < 03, l/2
-W(t)1
T/u,
(1.1)
ase
+ In, T])1’2
= t--i r:
1
a’s*
(1.2)
Here we set In u = log(u V e) and In, u = l&n u) for u > 0. Chen, Kong and Lin (1986) and Hanson and Russo (1983a, 1983b, 1989) have followed the same lines of investigation by considering statistics of the form given in (1.1)~(1.21, with the supremum over 10, T - aTI replaced by the supremum over smaller families of sub-intervals of 10, Tl. Furthermore, assuming suitable growth and regularity conditions imposed on uT, they proved that results of the type
Correspondence
to: Charles
0167-7152/93/$06.00
El-Nouty,
185 bvd Vincent
0 1993 - Elsevier
Science
Auriol,
Publishers
75013 Paris, France.
B.V. All rights reserved
27
Volume
17. Number
STATISTICS&PROBABILITY
1
3 May 1993
LETTERS
given in (l.l)-(1.2) were also valid for other functionals than that considered in these statements. We refer to Hanson and Russo (1989) and the references therein for further details and limit ourselves to the following result, commonly referred to as the Hanson-Russo law of the iterated logarithm. We have almost surely W(T) - W(T-t) lim inf T-CC
E [I,
sup +
(2t[ln T/t + In, t])12
w41
i =F(a)
whenever lim inf q/T T+-
whenever q-/T + a,
= a, (1.3)
where F(u)
= i
0
if a =O,
-1/{1-f(loga)}1’2
ifO
Let (B,(t), 0 < t < 1) be a fractional Brownian motion with index 0 < H < 1, i.e. a centered Gaussian process with stationary increments such that B,(O) = 0 a.s. and satisfying E(II,(c))~ = t2H (see e.g. Taqqu, 1977; Ortega, 1989; and the references therein for further details on the properties of BH). For H = i, {BH(t), 0 d t 6 1) so defined is a standard Wiener process. Denote by K, the unit ball of the reproducing kernel Hilbert space II pertaining to BH( *) and endowed with the norm II IIr. Taqqu and Czado (1985) have considered these spaces in the general framework of self-similar processes. Let C[O, 11 denote the set of continuous functions on [O, 11, endowed with the topology of uniform convergence. Whenever a sequence {f,, IZ> 1) of functions in C[O, 11 is relatively compact, we denote by _P(f,) the limit set of {f,}, i.e. the set of all limits of convergent subsequences Cf,,} of f,, The celebrated Strassen (1964) functional law of the iterated logarithm (LIL) shows that, if f,(t) = (2n In, n)-‘/2W(nt), then the sequence {f,, II z 1) is almost surely relatively compact with limit set .-Y(f,J = K1,2. Taqqu (1977) established a version of Strassen’s functional LIL for fractional Brownian motion. Namely he showed that BH(nt)
_!z
=K,
(1.4)
a.e.
nH(2 In, n)1’2 1
R&&z (1979) obtained a functional form of Strassen’s LIL, appropriate to (1.1). The main purpose of this paper is to prove a functional version of the Hanson-Russo LIL (1.3), in the spirit of R&&z (1979) for the fractional Brownian motion BH. Our main results are given in the following Section 2. The proofs of our theorems are postponed to Section 3.
2. Results To motivate further our statements, we first outline the arguments used by Taqqu (1977) to prove the LIL for BH. Towards the aim of proving (1.4), Taqqu (1977) showed in the first place by general methods that _Y(f,J c K, a.e. with f,(t) = BH(nt)/{nH(2 In n)i/‘}. Then, he achieved equality of these sets by showing that _Y(f,*) = K, a.e. for a suitably chosen subsequence. We will follow roughly the same arguments, with a different method for the inclusion -!Z’(f,) c KH. Towards proving this statement, we will use the results of Lai (1973, 1974) stated in Theorem A below: Theorem A. Let {X(t); 0 G t G l} be a separable centered Gaussian process with continuous covuriance function R(s, t 1 satisfying JW(t) 28
-x(4)2
Ogt
O
(2.1)
Volume 17, Number 1
STATISTICS&PROBABILITY
LETTERS
3 May 1993
where W is a continuous nondecreasing function on [O, 11 such that m
/1
‘Z’(e-‘)
du < +m.
Let {X,,(t >, n 2 l} be a sequence of Gaussian processes defined on the same probability space and having the same distribution as the process X(t), and let Y,(t)
= (2 In n)-1’2Xn(
t).
Then, with probability one, the sequence {Y,(t), n > 3} is relatively compact in C[O, 11 with set of limit points contained in the unit ball K of the reproducing kernel Hilbert space pertaining to R. Letting F,={Xi(t):Ogt~l,l~j
and
F=
fi F,, n=l
suppose furthermore
that F is a Gaussian family of random variables such that
n+m~~n_r,E(E(Xm(t)
I F,))2 = 0
for each t E [0, 11.
Then with probability one, the set of limit points of {Y,(t),
n 2 3) in C[O, 11 IYS equal to the set K.
Remark. It is readily verified that B,(t) follows the condition (2.1), since E(B,(t) Let g(u) be a positive nondecreasing function on [0, 03[. Set, for n 2 1,
X(f) =
BH(n
+
@b>) -BH(n)
swH
)
(2.2)
- B,(s))~
q
= ) t -s I 2H.
o
Our main results are stated in the following two theorems. Theorem 1. (i) Zf g(u) = o(ue) as u + 03 for all fuced p > 0, then, with probability one, the sequence ((2 In n)-1/2X,(t), n 2 1) is relatively compact in C[O, 11 and its set of limit points in C[O, l] is equal to K “iii) Zf g(u) = ua$( u ) as u -+ a for some 0 <(Y < 1, +(u) + (I)(U))-’ = o(ua> as u + 03 for all fixed p > 0 and if lim,, Ilim,_,,p”~(pu>/Jl(u> = 1, then, with probability one, the sequence {(2(1(Y>In n)-‘/2X n(t) 7 n > l] is relatively compact in C[O, l] with set of limit points in C[O, l] equal to K,. Theorem 2. (i) sequence {(2 In, (ii) Zf g(u) = sequence {(2(1 + contained in K,.
Zf g(u) = ~(1 + o(1)) as 1.4+ 03 for some 0
3. Proofs c K, a.e. and the sequence ((2 In n)-‘/‘X,(*)} is a.s. relatively compact in C[O, 11. To prove the reverse inclusion, we will use Theorem 4.1. of Mangano (1976) for a well-chosen subsequence. This result shows that Theorem Proof of Theorem 1. To prove (i), we first note that, by Theorem A, _Y((2 In n)-‘/2X,(*))
29
Volume
17, Number
1
STATISTICS
&PROBABILITY
LETTERS
3 May 1993
A remains true when condition (2.2) is replaced by a weaker condition of asymptotic independence. have here
vw)xnw)
=
2g(m)ig(n)H [(m-n+fg(m))2H+
We
lm-n-sg(n)12H
For m 2 n and g(m) = o(m - n) as n --f m, a second order expansion yields E(X,(t)X,(s))
-H(2H-
1)ts
M4s(411-” [(m -n)2]1-H
’
(3-l)
Choose any p > 1, and set nk = [ kJ’] for k = 1, 2.. . . Since g(u) = o(zL(~-~)/~) and nk+l - nk -pkpP1, we have g(n,+,) = o(nk+l - nk> as k + w. We will now show that for each k E N*, there exists an integer k, a k such that 2[kP]g[(k+r)P]
forall
rzk,-k.
(3.2)
Since r > k, -k implies k,P - 1 G [k,P] G [(k + rjP], we see that (3.2) holds whenever 2kP f k,P - 1, or equivalently k, > (2kP + l)‘lp. Next, we prove that for any integer k’ > k + r, we have nk’
-n,ar.
(3.3)
To prove (3.3), we start with the obvious inequalities (following from an application of the mean-value theorem to the function x +xp on [k, k’l) pkp-‘(
k’ - k) < ktp - kP
k’ - k).
Since k ” - kP G [k’P] - [kP] + 1 = nkt - nk + 1, we see from this inequality combined with k’ - k > r that (3.3) holds whenever k’ - k
k’ - k) - 1~ nkt - nk.
Finally, since k’ - k z 1, the latter inequalities are satisfied for k a (2/p)‘/(p-1). By (3.1), we see that for any E > 0, there exists an integer k, such that for each integer k a k, and for each integer k’ > k + r with r z k, - k,
(3.4) Observe that dnk’)dnk) (nkf
30
-nk)2
=p dnkf)
gbk)
nk’-nk
(nkr-nk)1’2
1 (nkt-nk)1’2’
STATISTICS&PROBABILITY LETTERS
Volume 17, Number 1
3 May 1993
nk). We deduce from (3.2) that 2n, G nkl and Since nkl -nk’-l & nk’ -nk we have g(n,,)=o(n,,consequently ny’ < (n,, - nk)l12. Since g(n,) = o
g(n&(nk)
~
1 (3.5)
cnk’
-nk)2
(nk’-nk)1’2’
By combining (3.31, (3.4) and (3.51, we see that for any E > 0, there exists an integer k, such that for each integer k > k, and for each integer k’ > k + r with r 2 k, - k,
_‘, k
and therefore
It readily follows that
Hence, by Theorem 4.1 of Mangano (1976), P((2 In k)-1/2Xn,(. k --) 03,we obtain that _~?((2 In n)-l/*X,(
e)) IP-~‘~K,
1) = K,
Lemma.
nk
N
P In k as
a.e.
But p > 1 is arbitrary, and so _Y((2 In n)- ‘12Xn( *)) = K, a.e. To prove (ii), we consider 1 < 6 < l/(1 - a) and mk = [k’l.
P((2 In k)-1/2X m!i(.)I cK
a.e. Since In
BY
Lai
(1%‘4),
we
have
H’
We have uniformly over mk < m < mk+ ,,
+
SUP O
2g(mk)-H
i&(m+&(m))
-BH(mk+tg(mk))i’
proof. Let t be fixed. By combining the triangular inequality and the fact that g is nondecreasing, follows that BH(m
+
@trn))
-hdrn)
BH(mk
+
gwH BH(mk
-BH(mk)
dmk)H +
@tmk))
-BH(mk)
<
-
g(m)” BH(m
&tmk))
it
+
@cm))
+ swH
Bh(mk
ffg(mk)) dmk)
-BH(m)
-
-‘fdmk) H
BH(mk+tg(mk))-BH(mk) g(m)”
31
Volume
STATISTICS&PROBABILITY
17, Number 1
l-
+
g(m,) ~ ( g(m)
H 11
BH( m + fg( m)) -BH(mk
+ ‘dmk>)
II
@klH
This proves the lemma.
3 May 1993
LETTERS
+
BH(m)
-BH(mk) dmkjH
’
0 k a 3) is relatively compact in
Proof of Theorem 1 (continued). Since the sequence ((2 In k)-‘/2X,k(*), C[O, 11 with probability one, it follows that sup II(2 In k) -“2xm,(
0) II<
m
a.e.
k>3
Moreover because g is nondecreasing
and since
we have
lim
(3.6)
k-m
mk + dmk+l) - dmk). Since mk+l - mk N 6ksm1 and g(m,) Let Pkcmk+lpk = o(g(mk)). We now need the following bounds.
P
sup
- k**$([k*]),
we have
h k)1’2 IBH(m+r(g(m))-BH( mk+tg(mk))l>,i?g(mk)H(2
mk
O
J
r
I&(t)-&(S)1 sup m,-g(m,+,)ds,t~mk+l-g(m&) lS--fl
=P
sup
1BH(t)
-hdS)
>cg(mk)H(21n
1 2
(Pk)
O~s,tgm&+,--m&+2g(mk+,)
HE
% [
is-ti
k)1’2
1 H(2
In
By Lemma 5 of Ortega (1989), the exponential term of the majoration is therefore
k)l12
I.
For E > 0, we can choose k so large that the above exponent is greater than one. Thus, we have lim k-tm
sup mk
O&t<1
32
g(mk)-H(2
In k)-1’2
1 BH(m
+ 6drn>)
-BH(
mk + tg(m,))
I= 0
a.e.
Volume 17, Number 1
STATISTICS&PROBABILITY
LETTERS
3 May 1993
Thus, by combining the lemma and (3.6), we obtain that lim
sup
k-too mk
(2 In k)-l’*
Since _Y((2 In k)-‘/*X,k(.))
c K,,
11X,( *) -Xmk( *) II= 0
we have c~-~/*K,
_5?((2 In n)-l/*X,,(.))
a.e.
a.e.
for any 1
Letting S 7 (1 - (~)-l, we so obtain that _Y((Z In n) -“*XJ
*)) c (1 - a)l’*KH.
To prove the reverse inclusion, we use a similar argument as that given for (i), making use of the fact that g(m) = o(m -n). We also choose the subsequence nk = [kPl for k & 1 such that p > (1 - a)-l. 0 Proof of Theorem 2. To prove (i), for any fixed E > 0, we choose c > 1 such that E*CZ*~> 4(2(c - 1X1 + CZ>>*~. Let also mk = [c kI. For mk
IBH(m+tg(m))--BH(m,+tg(mk))I sup l?Zk
>g(mk)Ha(21n
k)1’2
Ogta1
IBH(t) -BH(S)
sup 0~s,t~mk+,-mk+2g(m,+l Is-tlGPk [
&
=qP
h
k)l’* I
I BH(f)
sup v=l
1 ag(mk)Hs(2
)
-B&s)
I ag(mk)H@
h
k)“*
I b-l)Pk4srt<(v+l)p~ SUP
IBH(t)
-B,(S))
k)l’*
ag(mk)Hc(2h
[ Oe,t<2pk
I BH(t)
I ag(m,)“t(2
In k)“2
1.
1
1
By Ortega (1989), the exponential term of the majoration is therefore
But a
2(c-1)(1+(Y)
1 2H
0
‘l
~*~x*~>4(2(c-l)(l+a))*~.
Thus we have
lim sup k-m
SUP
[ g(mk)H(2
h
k)l’*lpl
I BH(m
+
e(m))
-BH(mk
+
@drnk>)
I GE.
m~
O
33
Volume
17, Number
STATISTICS&PROBABILITY
1
LETTERS
3 May 1993
a.e. By all this and the lemma, we obtain that lim sup k+-
sup m~
“x~(*)-xmx(.)” (2 In k)“2
< (‘“-l)
limsup(2In k+m
CH
k)-1’2((X
“k
(*)((+E
o
1)
(P-
G
SE. CH
By Theorem A, we have _‘5?((2In k)-‘/2Xm,!*)) cK, a.e. Since ln k N h2mk as k + ~0, we have _!Z((2 ln2m,)-‘/2Xmk(~)) c K,. Hence, since E > 0 can be chosen arbitrary small, we have
To prove (ii), take 6 = (1 -i-p)-‘. We choose c > 0 such that &2A2H> 4(2~a+‘6)~~ and mk = [exp(ck’)]. pk = mk+ 1 - mk + dmk+ 1) - dm,>, we have pk - ci3k-‘P exp(ck*). The proof follows along the same lines as the proof of (9. q htting
Acknowledgment
I am grateful to Professor P. Deheuvels for his helpful comments and also to the referee remarks that have led to improvements in this work.
for several
References Book, S.A. and T.R. Shore (19781, On large intervals in the CsiirgB-R&&z theorem on increments of a Wiener process, Z. Wuhrsch. Verrv. Gebiete 46, l-11. Chen G., F. Kong and Z. Lin (19861, Answers to some questions about increments of a Wiener process, Ann. Probab. 14, 1252-1261. CsiirgB M. and P. Rev&z (1979), How big are the increments of a Wiener process? Ann. Probab. 7, 731-737. Hanson D.L. and R.P. Russo (1983a), Some results on increments of the Wiener process with applications to lag sums of i.i.d. random variables, Ann. Probab. 11, 609-623. Hanson, D.L. and R.P. Russo (1983b), Some more results on increments of the Wiener process, Ann. Probab. 11, 10091015. Hanson D.L. and R.P. Russo (1989). Some “lim inf’ results for increments of a Wiener process, Ann. Probub. 17, 1063-1082. Lai T.L. (19731, On Strassen-type laws of the iterated logarithm for delayed averages of the Wiener process, Bull. Inst. Math. Acad. Sinica 1, 29-39.
34
Lai T.L. (19741, Reproducing Kernel Hilbert spaces and the LIL for Gaussian processes, Z. Wahrsch. Verw. Gebiete 29, 7-19. Mangano G.C. (19761, On Strassen-type laws of the iterated logarithm for Gaussian elements in abstract spaces, Z. Wuhrsch. Verw. Gebiete 36, 227-239. Ortega J. (1989), Upper classes for the increments of fractional Brownian motion, Probab. Theory. Rel. Fields 80, 365-379. Revesz P. (19791, A generalization of Strassen’s functional law of iterated logarithm, Z. Wahrsch. Verw. Gebiete 50, 257264. Strassen V. (1964), An invariance principle for the law of the iterated logarithm, Z. Wuhrsch. Verw. Gebiete 3, 211-226. Taqqu M.S. (1977), LIL for sums of non-linear functions of Gaussian variables that exhibit a long-range dependence, Z. Wuhrsch. Verw. Gebiete 40, 203-238. Taqqu M.S. and C. Czado (19851, A survey of functional laws of the iterated logarithm for self-similar processes, Comm. Statist.-Stochastic Models. 1, 77-115.