No. 1781
DYNAMICAL SYSTEMS. FRACTIONAL BROWNIAN ManON
AND LIMIT THEOREMS
•
Michael T. Lacey
University of North Carolina. Chapel Hill
AMS Subject Classification.
Primary:
6OF17
Secondary:
•
6OF15. 47A35
Key Words and Phrases: dynamical systems; functional central limit theorem;
functional law of the iterated logarithm; fractional Brownian motion;
reproducing kernel Hilbert sPaCe.
ABSTRACf
Let
2
(X.~.T)
be a dynamical system. and for f € L (X) set
Sn f
~
and for 0
t
~
=f
+ f
0
T + ... + f
S f(t)
n
(X.~.T)
={
t
-1
an
= IX(Sn f) 2~
= kin.
k
= O.I.2 ..... n
linear inbetween .
be aperiodic. and fix 0
< H < 2.
fractional Brownian motion of index H.
so that a n
2
~-1
1
•
1
~f
Let
0
Let
~(t).
(See [MvN])
0
~
t
~
1 be the
2
Then there is an f € L (X)
Sn f(·) converges weakly to -H
R_(.) in C[O.I].
result for the law of the iterated logarithm also holds.
The corresponding
Both results. by way
of a new construction. extend a central limit theorem due to Burton and Denker
[BD].
1.
Introduction
In an interesting paper, Burton and Denker [BD] established the central
limit theorem of probability theory in some dynamical systems.
result, let
(X,~,T)
measurable,
~-measure
2
f € L
(~)
Theorem 1.1.
-00
preserving map on X, and
be centered, IX f
2
2
an = fX(Sn f).
all
be a dynamical system with
~ =
~(X)
(X,~)
0, set Sn f = f + f
= 1.
To state the
That is, T is a
is a Lebesgue space.
0
T + ... + f
~-1
01
Let
,
and
Then
2
If (X,~,T) is aperiodic, then there is an f € L (X) so that for
< u < +00,
.
Sn f(x)
}
2
-a-- - ~ u .... (21r)~ I~ exp(-v /2)dv, as
n
(1.2)
(1.2) is a central limit theorem, briefly, CLT.
n .... +00.
Most CLT's are proved
under some restriction on the dependence structure of f. f
0
T.....
Thus. the
result above is most striking in the zero entropy case. where one expects the
most dependence.
We refer the reader to [BD] for more background on these
points. and note that the papers [Ma], [GS] and [CS] are related to Theorem
1.1.
For specific dynamical systems they find general sufficient conditions
for (1.2) to hold.
The CLT continues to be one of the most studied results in probability
theory, and has been the subject of many generalizations.
Theorem 1.1 clearly
raises the possibility that some of these generalizations can hold in the
context of dynamical systems; it is this point which we address in this paper.
Two fruitful generalizations in the case of independent. identically
- 2 -
distributed summands are due to Donsker [D] and Strassen [St].
These results
are. respectively. the functional central limit theorem and law of the iterated
logarithm.
Both results treat the limiting behavior of the following
continuous functions on t
[0.1].
€
For x
~ f
(1.3)
Sn f(t.x) =
X. and integers n. let
€
t = kin. k = 0.1.2 •.... m
{ linear inbetween
Note that S f(-.x) is a continuous function on [0.1].
n
Thus an
1
S f(-.x)
n
induces a probability measure on C[O.l]. the space of continuous functions on
[0.1]. with the usual sup-norm.
weakly towards a limit.
The FCLT states that these measures converge
More specifically. let Y(-). t € [0.1] be a stochastic
process with P(Y(-) € C[O.l]) = 1. 'Write f € CLT(Y) if for all bounded.
continuous h : C[O.l]
~(x €
(1.4)
~
m.
X : h(a
-1
n
and all
-00
< u < +00.
....:
S f(-.x»
n
< u)
~
j
-
'
P(h(Y(-»
~1-
< u).
n
~
+00.
This is nothing more than the definition of weak convergence of probability
measures on the separable metric space C[O.l].
(See e.g. [B]).
We also note
the FCLT is a statement about the statistical regularity of the evolution of
Sn f.
Donsker's theorem can be summarized as follows.
Brownian motion.
(See [B].)
Let B(t) be a standard
For independent identically distributed random
variables. with f f=O and f f2=1. f € CLT(B).
The class of processes we consider include the Brownian motion as a
special case.
2. let
~(t).
They are the fractional Brownian motions.
t
>0
be a Gaussian process which satisfies
~(O)
and for s. t
(1.5)
= O.
E ~(t)
= O.
t
> 0;
> O.
E(~(s+t) - ~(t»2 =
H
s .
([MvN]).
For 0
<H ~
- 3 ~
For a construction of these processes, in particular, a proof that
admits a
version with a.s. continuous paths, we refer the reader to [Ka, Chapter 18].
We further comment that
has a self-similarity property:
lV2
{a~(at)
(l.6)
(g
~
> O} g {~(t)
t
: t
is equality in finite dimensional distributions.)
For all a
>0
> O}
Finally, observe that for
H=2, B (s) can be realized as s·g where g is a standard Gaussian random
2
variable (mean 0, variance 1). That is B is a trivial process.
2
Theorem 1.7.
Fix 0
< H < 2.
For all aperiodic dYnamical systems
(X,~,T),
2
there is an f € L (X) for which f € CLT(~).
By a theorem of Lamperti, [L, Theorem 2, p. 64], [M, Theorem 1], if f €
CLT(Y) for some stochastic process yet), then yet) is self-similar.
yet) satisfies (1.5) for some H
> O.
That is,
Moreover, if yet) is square-integrable,
J.',
then
0
2
n
= E y2 (1)
stationarity of f
• n
0
(1.4) is satisfied.
H
~,
L(n): where L(n) is slowly varying.
k
~ O,~{~)
Further, by the
must have stationary increments, so that
Finally, if yet) is Gaussian, it must be a version of
Thus, the fractional Brownian motions,
~
for 0
< H < 2,
~.
are the largest class
of non-trivial Gaussian processes which can arise in Theorem 1.7.
The LIL can be thought of as an almost everywhere version of the CLT. To
2
2.' + ' . ; , + 2
state the FLIL, set a = 2 0 log (1 + log 0). For a compact subset K of
n
n
n
C[O,I], write f
x
~
€
FLIL(K) if there is a set N C X with
X\N, the following two conditions hold.
~
C
For
~(x)
~(N)
= {an-1
.
C[O,I],
( l.8.a)
~(x)
is relatively compact in C[O,I];
and
(l.8.b)
the set of cluster points of
~(x)
is K.
=0
so that for all
Sn f(·,x) : n
~
I}
- 4 For each 0
< H < 2 there is a compact KH
C C[O.l] which is the unit ball
with~.
of the reproducing kernel Hilbert space associated
(See [JM].)
H=l. K was identified by Strassen. [S].
1
t
1 2
K = {F € C[O.l] : F(t) = f O f(s)ds. where f O f (s)ds
1
~
For
I}.
For H # 1. there is a description of KH in terms of fractional integration.
[TC1]. indeed Mandelbrot first defined
~
by fractional integrals. hence the
name.
Theorem 1.9.
Fix 0
< H < 2.
For all aperiodic dYnamical systems
(X.~.T).
2
there is an f € L (X) with f € FLIL(KH).
Consider the projection g
~
g(l) on C[0.1].
Then as a corollary to this
theorem. we get
lim sup a- 1 S f = sup g(l) = 1
n
n
n
g€KH .
a. e. (X)
This is the usual law of the iterated logarithm.
2
2
The set of f in L (X) satisfying any of these theorems are dense in LO(X)
2
= {g € L (X) : fxg = O}.
This is easiest to see from the fact that {g - goT
g € L2(X)} is dense in L~(X). and an
191] and [GL Remark 1. p. 393].)
= O(nH).
~ (See [B. proposi tion 2.2. p.
The classes of f which satisfy Theorems 1.7
and 1.9. for which the rate of convergence in the pointwise ergodic theorem is
very regular. contrast with the classes constructed by Kakutani and Petersen.
[P, Theorem 2.3. p. 94]. for which the rate is bad.
Other questions about the
nature of these classes. such as first or second category. are open.
Also. there is a broad class of processes yet) satisfying (1.5) for some H
> O.
Besides stable processes which are not square-integrable. there are
non-Gaussian self-similar processes which are square integrable.
It would be
of interest to extend theorems 1.6 and 1.S to any of these processes.
(See
- 5 -
[TC2], [M] for surveys of limit theory and self-similar processes.
Also see
the bibliography [T].)
We now comment on the proof of theorems 1.7 and 1.9, and the organization
of this paper.
The constructions of the functions f which satisfy these
Theorems are very different from those in [BD].
(This seems to be necessary.)
In section 2, we treat the case of irrational rotation.
For fixed 0
the function f is constructed in terms of it's Fourier expansion.
< H < 2,
Then
classical facts about lacunary trigonometric series are used to give a complete
proof of Theorem 1.7.
We do this to avoid carrying out very similar
calculations in the technically more complicated section 3, in which the
general case is taken up.
Moreover, many aspects of section 2 are closely
mimicked in section 3, thus section 2 should be read before section 3.
full proof of Theorem 1.7 is contained in section 3.
~
The
The extra steps needed to
prove Theorem 1.9 are described ,in section 4.
Acknowledgements
We thank Manfred Denker for first suggesting to us that theorem 1.6 could
be true.
Our thanks also go to Karl Petersen for rekindling our interest in
these theorems, and patiently explaining some well-known facts.
2.
Irrational Rotation.
irrational with respect
(2.1)
Then observe that
K (A)
n
and let T x = x+a mod 2~ where a is
. ijx
Write f(x) = ~ bje
, where ~ b~ < +ro, and
Let X
to~.
=1
= R/2T,
+ e iA + ... + e i (n-1)A
J
1 - e
inA
= ---~~
1
iA
- e
- 6 -
S
n
f(x) = f(x) + f
0
T(x) + ... + f
rn- 1 (x)
0
=! b. K (j a)e
J
j
ijA
n
.
We will choose the Fourier coefficients to
+00
satisfy in addition. b
Thus f(x) = 2!
= 0 and b j = b_ j .
O
1
fact. f will be a sum of lacunary trigonometric functions.
b j cos j x.
In
Thus. the proof
will use well-known properties of such functions. e.g. the Salem-Zygmund CLT
and exponential-squared integrability.
The next lemma gives the construction of f:
Fix 0
Lemma 2.2.
< H < 2.
There is a set J = {jl. j 2 •... } eN and constants b j
so that
r=I.2.3 .... ;
(2.2.a)
(2.2.b)
. if
(2.2.c)
Ij I
t J;
b~ < +00;
!
J
2
a 2 =! b21 K (j a) 1
(2.2.d)
n
j
H
= n
n
+ O(log n)
as n -+
+00;
and
max a -1 b
(2.2.e)
n
j
j
IK (j
n
a)
I = O(nfV2 )
as n -+
+00 •
Conditions (a) and (b) guarantee lacunarity; (c) implies that f
€
2
L (X);
2
(d) shows that the L -norm of S f has the correct behavior; and concerning (e).
n
note that S (f) is the row-wise sum of the array of functions
n
{2 b r Re(Kn (j r
a»
cos j r x} r.n~'1.
Then (e) implies that this array is row-wise "infintesinal" (S f is the sum of
n
many small pieces).
Such conditions are frequently imposed to obtain a CLT.
- 7 -
In particular. the CLT 0-1 8 f ~ N{O.l}
n
is immediate from {the array version
n
of} the Salen-Zygmund CLT.
{[Z. Vol. II. Theorem 5.4. p. 264]}.
The next lemma completes the proof of Theorem 1.6 in this special case.
{2.3.a} below shows that the finite dimensional distributions of 0-1 8 f{t}. 0
n
n
~
t
~
1 converge to those of
functions 0-1 8 f{t}.
n
~{t}.
That is. the finite dimensional distributions determine
n
the distribution of 0-1
8n f{t}.
n
Lemma 2.3.
The following two results hold.
For all 0
~
t
and for all 0
> O.
n
> 2/0
{2.3.b}
I{x € m/2v:
1
< t
2
< ... < t
2
and X
In particular. for all X
>C
r
~
1. as n
oH.
lim I{x € m/2v:
Lemma 2.4.
2
n
sup
18 f{s} - 8 {t}1
/s-tl<o
n
n
> X a }I
n
=
o.
Lemma 2.2 proceeds by
by an appropriate integral. which we introduce now.
Fix 0 < H < 2.
There is a real-valued. sYmmetric. integrable
function w{x} on [-v.v] so that
{2.4.a}
n
2
X
C 0- 1 exp{- ----).
C oH
We now turn to the proof of two lemmas above.
0
> X a }I
>0
010
approximating
~ +00.
sup
18 f{s.x} - 8 f{t.x}1
Is-tl<o
n
n
~
{2.3.c}
This and the a.s. continuity of -H
Fl_{t} imply
For a proof of this. see [B. Theorem 8.1. p. 54].
Theorem 1.6.
{2.3.a}
{2.3.c} below shows "tightness" of the
- 8 -
and
Proof.
o.
w(x) = O( lxiI-H). Ixl !
(2.4.b)
For H=I. we can take w == 1.
Take H '# 1; assume for the moment that
A
w(x) exists.
A
Then by sYmmetry. wen) = w(-n).
Moreover.
I I(x) 12 - IK (x) 12 )w(x)dx
(n+l) H - nH = I :II"
(K
-'lr
n+
n
=
f'" ( e - jnx +
-v
A
A
A
= w(O) + 2w(I) + ... + 2w(n).
A
Setting K = O. we then see that w(O) = 1.
O
A
(2.5)
wen) =
=
~«n+I)
~
H
For n
~
H
1.
H
- 2n + (n-I) )
H-2
H-3
H(H-I)n
+ O(n
). n
~ +00.
The remainder of the proof consists of showing that a function with
Fourier coefficients as in (2.5) is integrable and satisfies (2.4.b).
Consider the case 1
< H < 2.
g(x) =
Then
~
nH-2 cos nx
n>I
~
x I-H f(3-H) sin(v(H-2)/2)
and the sum converges uniformly on Ixl
186].)
>~
as
for all ~
Thus g is integrable. and satisfies (2.4.b).
observe that by (2.5). w(x) -
~
x
I
~
> O.
O.
(See [7. Vol. I. p.
To conclude the proof.
H(H-l)g(x) is a continuous. bounded function on
[-v.v].
For the case 0
< H < 1.
note that the Fourier eXPansion for w(x) converges
uniformly on [-v.v) to a continuous function. hence w(x) is integrable.
- 9 -
Concerning (2.4.b). note that w(O) = O. but w(x) is non-negative on an
neighborhood of O.
below.)
(This follows from continuity of w. (2.4.a) and (2.7)
Nonnegativity ·and integrability are the only facts used in the proof
of lemma 2.2.
Nevertheless. to establish (2.4.b). from [Z. Vol. I. p. 186].
h(x) = ~ nH- 1 sin nx
n)O
~ xH- 2 f(3-H) cos v(H/2)/2
Now.
~
w'(x) = -2
"
n w(n) sin nx.
n)O
Thus. by (2.5). w'(x) - H(H-l)h(x) is a bounded. continuous function.
This
proves (2.4. b).
•
Proof of lemma 2.1.
Fix 0
< H < 2.
and w(x) as in lemma 2.4.
non-negative on a non-empty interval [O.a).
union of decreasing intervals I
r
Then w(x) is
Write this interval as a disjoint
with midpoints i • for r € rn.
r
Further.
require that for all r
(2.6)
Inductively choose J = {jl' j2.···} ern as follows.
(Recall that Tx = x + a).
jr+l a € I r +1 ·
Then set b j
Having chosen jr €
=0
for j (
= II
J.
bj
J. r
Pick jl so that jl
~
= b_ j
a €
1. pick jr+l ) 4jr so that
and
w(x)dx. r ~ 1.
r
Thus (a) and (b) hold. and (c) is satisfied as w(x) is integrable.
To check the last two conditions of the lemma. we use the following two
properties of K.
n
For all n.
II·
- 10 -
(2.7)
and
(2.8)
IKn (x) - Kn {y)1
~ n2 Ix-YI.
Then
= Ilxl>a IKn {x)1
2
+ 2 C sup{IK {x)1
n
~
2 : Ixl > {8 log n)-I}
2
+ C sup{ sup
r
Iw{x)ldx
x,y€I
IKn (x) - Kn (y) 1 : i r < (8 log n)
-1
}
r
0(1) + O{log n) + 0(1), n
~ +00.
Here, the first and second estimates follow from the integrability of w{x) and
(2.7).
The third follows from (2.6) and (2.8).
have shown a little more that (c) above:
For use later, we note that we
For all n,
(2.9)
< Clog n.
(Recall that the intervals I
r
It remains to prove (d).
are decreasing.)
By (2.4.b), (2.6) and (2.7)
sup sup Ib r Kn (j r a)1 2
r
n
~
-1 <
C sup x-1-H exp{-x)
O<x<a
+00.
•
- 11 -
Proof of lemma 2.3.
We apply the Salem-Zygmund CLT [Z, Vol. II, Theorem 5.4,
p. 264], to prove (a).
~
Fix 0
t
< t 2 < ... < t r
1
~
1.
It is enough to show
that for all a , ... ,a
1
v
•
(2.10)
G
n
= an-1
v
d v
~ a
S f(t ) ~ ~ R_(t ), as n
1 u n U l -H u
~ +00.
By (2.2.d),
as n
Moreover, G(x)
=~
r
g rn cos(j r k), with
max
r
Igrn I ~ 0,
~
+00.
as n
~
+00.
Therefore
(2.10) holds.
Clearly (c) follows from (b), and (b) is in turn a consequence of
well-known techniques from the theory of empirical processes, see e.g. [B].
Lacunarity is again used:
(2.11)
For all constants a , r
r
1{lxl
<v
:
I~
r
We shall prove (c) with n
d.
= 2v
a r cos(j r x)1
and 6
= 2-d
~
1 and A
> 0,
> A}I
for integers v and d, with v
By (2.2.d) and the definition (1.3), this yields (c) as stated, with a
slightly worse constant.
Again by (1.3), for each n observe that S f(t) has
n
local extrema at the nodes t = kin, k = 0,1,2, ... ,n.
and the dyadic decomposition of the integers one has
sup
Is-t 1<6
Is
n
f(6) - S f(t)1
n
Using this observation,
>
- 12 -
<
v
!
Is
su~
k=d-1 i~2K
f-
i2v - k
S
fl
(i+1)2v - k
Also observe that
v
!
2H(d-k)/4
< C.
k=d-1
for all v
> d.
Thus it is enough to bound the expression below.
To do so we
will use stationary and (2.11):
v
!
su~
k=d-1 i<2
fJ. (x
Is
-k f(x) - S
-k f(x)
i2v
(i+1)v
v
>C Xa
!
I
2H(d-k)/4)
n k=d-t'
v
~
Is
!
k=d-1
v-k f(x)1
> C X an 2H(d-k)-4)
2
k
~ C
!
2k exp(-C X2 2H(d+k)/2)
1 k=d-1
2
~
3.
Towers.
-1 exp(-C X2 6-H ).
2
•
C 6
In this section we construct the functions f used to prove
Theorems 1.7 and 1.9.
Their relevant properties are collected in lemma 3.9;
indeed. Theorem 1.7 is an immediate corollary to this lemma. see the comments
following lenuna 3.9.
The construction uses Kakutani-Rochlin towers. which we
define now.
Fix an aperiodic dynamical system (X. fJ.. T).
Defini tion.
For a posi tive integer N and
Eo
> O.
call a measurable set F C X an
- 13 -
(N,~) tower if F, TF, ... , ~-1 F are disjoint, and ~(R) < ~ where R
N-l
U
= X\
o
TiF is the residual set associated with F.
The Kakutani-Rochlin lemma states that a tower exists for each pair
see e.g. [P, lemma 4.7, p. 48].
property:
Set k(x)
= inf{k
(N,~),
We also note that towers have an additional
: ~x € F}, for x € R, then ~(x) x € F.
This
implies that the orbit of any point in X can only enter the tower at the bottom
level.
Definition.
Let L and N be positive integers,
residual set R.
Fi x 0
Call a function g : X
< ~~ < v
~
~
>0
and F a
so t ha t e iA.IS an Nt h root
C speciaL for
(L,N,~,F,A)
(LN,~)
·
0 funIty:
tower with
e iNA
if the following conditions
hold.
.-
is equidistributed on
(3.1.a)
{I, e
iA
i
ii A
geT x) = e
g(x), x € F, 0
(3.l.b)
~
i
< NL;
and
(3.l.c)
g(x)
Recall the definition of K
n
Lemma 3.2.
= 0,
x € R.
in (2.1) above.
Let g be special for (L, N,
~,
F, A).
Then
(3.2.a)
(3.2.b)
S (g)(x)
n
= g(x)
(L-l)N
K (A), for x €
n
i
T F, 0
U
0
~
n
~
N;
and
(3.2.c)
I i
T F
Proof.
g ~
= 0,
0 ~ i
< LN.
(b) and (c) follow immediately from the definitions of g and K .
To see (a), observe that for x € F and 0
n
<n
= 1.
~
LN,
-14-
Isn g(x) I
by (2.7).
= O.
Moreover. SLN g(x)
= IKn (g(x»I
~ c/IAI.
Using this. and the fact that orbits only
•
enter tower at the bottom level. (a) follows.
Our function will be a sum of special functions. defined appropriately.
To do this we need the next lemma and the notation which follows.
Lemma 3.3.
Let F be an
(LN.~)
tower. and let v be any finite partition of X.
Then there is a finite partition
P of
F so that {T
~
P:
0
~ ~
< LN}.
a partition
of X\R. refines v on X\R.
Proof.
Consider the partitions of F
Then let P be a common refinement of P •...• P - .
1
LN 1
This concludes the proof .
•
Fix 0
< H < 2.
•
and let w{x) be as in lemma 2.4.
an interval [O.a). with a
decreasing intervals A . r
r
> O.
~
Then w{x) is positive on
Write this interval is as a disjoint union of
1. with rational midpoints Ar so that
(3.4.a)
Set
2
br
(3.4.b)
= fA
w(x)dx.
r
Choose an increasing sequence of integers N so that
r
(3.4. c)
Nr
> er
inf{r : N
r
and
~
exp(i Nr A)
r
= 1.
m}; then for all n.
- 15 -
r
(3.4.d)
Choose L
r
o < log n.
to be an increasing sequence of integers, and
~
r
> 0 so that
(3.4.e)
~
For r -> I, let Fr be an (L r Nr ,
r ) tower with residual set Rr .
Inductively define special functions gr as follows.
Given g r- I' let
T
r-
Let gl be special for (L ,
1
1 be the finite partition of X generated
(3.5.a)
~
s = r-1, r; 0
-e
Let
T
~
r- 1 be the sigmafield generated by
= T r- 1;
r 1 .
T
r- l'
Apply lemma 3.3 with F = Fr . and
let Pr be a partition of Fr given by the conclusion of the lemma.
Then. let g
(3.5.b)
l ~ N-
r
be special for (L , N ,
r
r
~
r
, A ) so that for all A
r
grlA is equidistributed on {e
il A
r
0 ~ l
€
Pr ,
< Nr }.
ClO
(3.5.c)
f =
:I b
-ClO
r
g .
r
This is the function which proves the Theorems.- For purposes of the proof, we
further define for all n,
(3.5.d)
Observe that for
fn
=
:I
Irl>ro(n)
b
g .
r
r
- 16 -
(3.5.e)
e •
r
and we have that e
r
is': measurable.
r
The next lemma contains the relevant properties of the broken-line
functions S f{t).
n
Of it's four parts. the last is a CLT with rate for the
finite dimensional distributions of a-I S fee).
n
n
below.)
(More remarks on the lemma are
To state it succiently. we will use the following notation.
integer v.
For!
= (a1 ..... a v )
€
Fix an
mU • with Ilaul ~ 1.
with Iku 1 ~ n. define
~(y;a.k)
'" '"
= ~(x € X
e'
and
Also let
T
2
be the variance of the last sum. which is explicitly given in
(3.14) below.
a
~
t
~
By way of explanation. note that if 6{t) denotes a point mass at
1. then h = I
than 1.
~
6{ku/n) is a linear functional on C[O.l] of norm less
Thus ~(e) above is the distribution of h{an
for pee).
1
S f (e»
n n
(d) below gives an estimate for the rate at which
pee).
Lemma 3.6.
(3.6.a)
The following properties hold for all n.
sup
k~n
1&-k{f-f n ){x)1
~ clog n. all x € X;
and similarly
~(e)
converges to
- 17 -
Ian2
(3.6.b)
(3.6.c)
for all 6
> 2In.
~{x:
~
>C
H
6 •
2
and X
sup
Is-t 1<6
~
- nHI
Is
c(log n) 2 ;
f (s.x) - S f (t.x)1
n
n n
n
> a X}
n
2
2 n -8 + C 6-1 exp(-C2 6-H X);
l
and
(3.6.d)
for all! and!. (which depend on v)
Dn
= supl~(y;!.!)
_ Cv
<
--
- p(y;!.!)I
y
T -415( 1og n )215 n -2Hl5 .
Observe that by (a) and (b) imply that
lim sup a-liS (f-f )(t)1
n-+f-ClO t
n
n
n
Thus. one only needs to consider S
f
n
n
=0
a.e.(X).
in proving Theorems 1.7 and 1.9.
(c)
shows tightness of the probability measures of a-I S f on C[O.l]. and (d)
n
n n
shows weak convergence to
Bn
for a dense set of linear functionals on C[O.l].
Together. they imply the functional central limit theorem.
For a proof of
this. see e.g. [B. Theorem 8.1. p. 54].
Proof.
Fix an integer n.
To see (a) observe that for all k
~
c (log n) max
r
~
clog n.
~
n.
- 18 -
Here. we have used (3.2.a), (3.4.a) and (3.4.d).
For the remainder of the proof. martingales playa crucial role.
background on martingales. see [D].)
sigmafields
~r'
(3.5.a)
abov~.
Recall the construction of the
We see that they are increasing in r.
E(·I~ ) denote conditional expectation (on X!) with respect to ~.
r
r
k
~
n.
By
Fix 0 ~ j <
Then. mimicking (3.5.e). write
=
~
~
e (j.k.n) =
r>r
r
r>r
O
Then e r is ~ r measurable. and setting d r
~r'
(For
= er
e.
r
O
- E(e r I~r- 1)' we see that (d r .
r > r O) is a martingale difference sequence (quickly. mds).
e"
Let
(3.8)
Gr
=
and Br = X\G r
(Recall (3.4.e). which controls the size of B .)
r
The following conditions hold.
x
(3.9.a)
(3.9.b)
sup
x
sup
j<k~n
€
Gr . 0
~
j
<k
~
n:
sup Id I ~ 2 sup sup suple I = 0(1). n ~ +00:
r
r
r>r
O
and
(3.9.c)
~{x
max
O~j<k~n
l(k-j)H -
~
r>r O
E(d~ l~r-1)1 > clog n} < n-8 .
We now show that (3.9) implies (b) and (c) of the lemma.
that by (3.6.a)
To see (b). note
- 19 -
2
2
= O«log
n} } + IX(Sn f}
n ~
= O«log
n} 2 } + I x (! E(e r I~r-l}} 2~
r
+ 2 I X(! E(erl~r_l}}(! d r }
r
r
Thus (b) will follow from the next two estimates.
First, by (3.2.a), (3.4.e)
and (3.g.a)
(Ix(!
r
E(erl~r_l)}2~}~
(3.10)
•
~
-8
en.
Second, by (3.9.c) and orthogonality of martingale differences, observe that,
= IX
!
r
= nH +
E(d~l~r_l}~
-6
O«n
!
r~ro
2
2
b}}
+ O«log n} }
r
•
= nH +
Here, we've used! b2
r
~
2
O«log n} }.
I 1r-:rr Iw(x} Idx < +00.
Then use Schwartz's inequality to
- 20 -
control the middle term above.
Part (c) of the lemma follows from (3.9.b) and (c) and an argument very
similar to that used to prove (2.3.b) above.
The only missing ingredient is an
exponential squared distribution inequality, like (2.11) above.
can use Bernstein's inequality for martingale mds.
299].
For this, one
Use [S, Corollary 5.4.1, p.
(For comparison, Bernstein's inequality for independent r.v. can also be
found in [S].)
~r-l
To prove (3.g.a), observe that by (3.5.a), Gr is
measurable.
if A eGis any ~ 1 atom, then A C T~ F for some 0 ~ ~ < (L -1}N.
r
rr
r
r
A is completely contained in a level of the rth tower.
Hence,
That is
Moreover, as r > ro(m) ,
~+k
_k
we have N > m. Therefore for 0 ~ k ~ m, we have r- ACT
Fr' which is a
r
level of the tower. Then the definition of g and ~ «3.5.b) above} show that
r
fA g
Thus E(e r I~ r- 1}(x}
=0 for x
rl<
0
€
dJ.L
= 0,
0
~
r
k
~ m.
e·
Gr .
To see (3.9.b), by (3.2.a),
sup
x
~
sup
O~j<k~n
C sup
r~rO
sup Ie (x) I
r
r>r
O
-1
b r Ar
In proving (3.9.c) we treat the case j
similar.
(3.11)
= O(I},
= 0,
n
~ +00.
the case 0 < j < k
~
n being
We claim that
for x €
n
r>r
G.
r
O
This proves (3.9.c): use (3.g.a) and (3.4.e) to see that ~(n
r>r
G) > I-m-S .
r
O
- 21 -
To prove (3.11), for x € G we have
r
e r (x)
=br
R {g
-k
r + g -r )(x)
= b r {gr (x)
~
Then for an atom A of
r-
K (X r ) + g r (x) -K
K (X r
-K
».
l' with A C Gr ,
12 .
fA e 2r (x) = 2 b 2r 1-K
K (X)
r
Hence, for x €
G,
r
Therefore, (3.11) follows from (2.9), (3.4.a), and (3.4.b).
It remains to prove part (d) of the lemma.
Fix
~
and
~
as in (d).
Then
write
a
-1
n
And by a further abuse of notation, set d r
= er
linear combination of mds, all with respect to
- E{e r I~ r- 1).
~
r
Then, d r is a
, hence it is a mds.
observation allows us to use a martingale CLT with a rate to prove (d).
This
Set
-22as ~Iau I ~ 1.
Let
L
n
= Ln"''''
(a,k) =
~
r~ro
IX
d
4
r
$.
and
As a consequence of the main result of [HB]. (also see [H]. Theorem 1)
applied with 6 = 1 and to
T-
1 e , we see that the 2.h.s. of (d) is dominated by
r
(3.12)
Our task is then to estimate this last quantity.
Observe that by (3.4.c).
e
(3.13)
+ n
4
~
r~rO
Secondly, to bound N • write
n
(3.14)
b
4
r
~(B»
r
w
- 23 -
and likewise for E(e21~
1).
r r-
Then by (3.9.c) we have
•
on a set of measure at least 1-n-8 .
Thus. Nn
4.
~
FLIL.
-2
2
C an (log n).
•
This. (3.13) and (3.14) finish the proof.
For the proof of Theorem 1.9. we will use the following notation.
Let Z(n.t.x) =
k
= [a].
Off of this set we have
a~l Sn fn(t.x); ~(n)
(Recall that a n
= an
~.)
n
= (2 log log
a~)~; for
a
> 1.
k
~ 1. let ~
We need to prove both parts of (1.8).
(For
a general discussion of our approach to this theorem. in particular our
reduction to (4.2) below. we refer the reader to [Ph. section 2].)
(1.8.a) follows from the Arzela-Ascoli theorem. and (3.G.c) as we now
show.
Let
~(~)
( 4.1)
To see that
observe that
~(x
~(x)
= (x
sup
Is-tl~a-1
IZ(~.s.x) - Z(~.t.x)1
is bounded in C[O.l])
=1
(~(x)
is defined in (1.8.a).).
- 24 sup
sup IZ(n,t)1
2k-1~n<2k t
C(k a-1k + sup IZ(2k ,t) I ).
~
2
t
Moreover, setting a = 2 in (4.1), we have
~ ~(x
k
sUPIZ(2 ,t,x)I
k
for e sufficiently large, by (3.6.c).
that
~(x
:
~(x)
> e ~(2k»
t
is bounded in C[O,l])
The Borel-cantelli lemma then implies
= 1.
A very similar argument will show that
~(x
~(x)
is equicontinuous)
= 1.
Observe that
sup
sup
IZ(n,s) - Z(n,t)1
~_l~n~~ Is-tl~a-1
where Ca ~ 1 as a
!
1.
Further, for all e
sufficiently close to I, by (3.6.c).
> 0,
~ ~(~(e»
< +m for a
The proof is then completed by the
Borel-cantelli lemma.
We turn to the proof of (l.S.b), with K
reproducing kernel Hilbert space
spaces.)
of~.
(See
the unit ball of the
[JM] for background on these
By applyiny [Kb theorem 3.1, and lemma 2.1 (iv)], we have the
following sufficient condition for (1.7.b):
(4.2)
=~,
lim sup
n
-1
~n
g(Z(n»
For all g € C[O,l] * ,
= (E g(~»
2
~
a.e.(~).
- 25 -
(Kuelbs' result has been used by several other authors in the context of FLIL
associated with fractional Brownian motion.
•
See the survey [TC2].)
Moreover.
by relative compactness. we can. and do. take g in (4.2) to be a finite linear
v
•
combination of point masses on the interval [0.1].
~Iaul
= 1 and T
2
= E
That is. g
=~
1
a
u
0
t
• with
u
g(~)2. This reduction will allow us to use (3.6.d) in
the argument below.
The upper half of (4.2) follows from relative compactness and
-1
~
lim sup
(~)g(Z(~»
~ T
a.e.(~)
k
for all
a
> 1.
To establish this limit. fix
> o.
~
As g is fixed. (3.6.d)
implies that
~(x
g(Z(~»
:
< (1+~)
T
~(~»
~ C T- 1 k2/5 ~2Hl5 + P(g(~) < (1+~)T ~(~»
~
CT
-1
which is summable in k. for all
2/5
k
~
> O.
-2Hl5
~
+
exp(-(I+~)
2
2
~ (~».
This concludes the proof of the upper
half.
To prove the lower half of (4.2). we will show that
a.e.
Fix
.
>0
~
and set
that
~(~
are:
~ ~(~)
i.o.)
=
~
= 1.
+CO
and
= (x
k
g(Z(2 .x»
~
k
~ (1-~) T ~(2
».
(~).
It is enough to show
By a result in [ER]. two sufficient conditions for this
- 26 -
{4.4}
•
In
The first condition follows from an argument very close to {4.3}.
fact. letting -#I{u}
= {2Jr}~
1+0»
u
•
2
e-v /2 dv. observe that
{4.5}
and
{4.6}
v
Verifying {4.4} will be more delicate.
Recall that g =
finite linear combination of point masses. with norm ~Ia
u
~
1
I = 1.
a
u
6{t} is a
u
....
Let g be any
other norm 1 linear functional which is a combination of at most v point
masses.
....
We have in mind g
= gk = ~
-k
a u 6{2
r.h.s. of {3.6.d} depends only on v and
two-dimensional a..T. with a rate:
Set
t}.
u
E
....2
T
= E
As the constant on the
g{~}
....
2
. we have the following
2
g{~}.
Then
sup I~{x : g{Z{m}} ~ y; ~{Z{m}} ~ z}
y.z
{4.8}
-
P{g{~} ~
~
C {T /\ T}
y;
~{~} ~
.... -4/5
v
z}1
{log m}
2/5
-2lV5
m
.
2
2
Set T k = E gk{~} • and observe that by self-similarity. {1.6}.
2
2-kHT . Then for j. k > 0
T
2
k
=
- 27 -
•
~
(4.9)
m(x
g(Z(2
j+k
»
> (1-~) ~T
~(2
j+k
);
where
6(j.k)
= 2-1dV2
sup IS k+j f k+j - S k+j f j
t
2
2
2
2
I
(4.10)
~ C«log k)2-1dV2 + Ij+kl 2-(k+ j )Hl2).
Here. we've used (3.6.a) and (3.6.b).
We will also need the following lemma on
Lemma 4.11.
~.
For all intergers j and k.
A (j.k) = Ip(g(~)
1
- P(g(~)
> (1-~)~T ~(2j+k); gk(~) > (l-~)~Tk ~(2j»
> (1-~)~T ~(2j+k»p(gk(~) > (l-~)~Tk ~(2j»1
j+k 2
j 2
) + ~(2 ) ) )
2(1+C 2-kG )
~ C 2-kG exp{- (1-~) (~{2
where G = min(Hl2. (2-H)/2).
Proof.
normal random variables. with covariance
..
Then Y and Yk are standard
-28IE Y Ykl
~
T-
2 2+kH/2
I
k
I
1~u,w~u
~ ~ T-2 2+kH/2
)1
E B(t )B(2- t
u
w
I
1~u,w~v
With this observation, the necessary estimate can be carried out in a
straightforward way.
See e.g. the Normal Comparison lenuna, [LLR, Theorem
4.2.1, p. 81].
•
We now have all the estimates necessary to prove (4.4).
We need to show
that
n
I
I
~(A
I
~(Aj»
A
)
j=1 1~k~n-j
j j+k
I im sup .......--=-=------:2:::----
(4.12)
n
(
1~Hn
~ ~
.
Here, we've used (4.6) to eliminate the diagonal for (4.4).
RO = {(j,k)
R = {(j,k)
1
1
1 ~ j
~
j
< n,
< n,
1
~
k
~
Let
(n-j)}
1 ~ k ~ (n-j) /\ n~/2}
and
~
= {(j,k) : 1
~
j
< n, 1
~
k
< j 2/~ /\ (n-j)}.
Notice that R is the set of indices over which we sum; R is "close" to the
O
1
diagonal of the square; and R and
1
~
overlap.
(4.12) will follow from the calculations below, which can be routinely
verified.
By (4.5) and (4.6),
- 29 n
~ L
(4.13)
L.
-R
Jl
1
(A
A
) / n~/2 ~ I.(A.) = 0(n3 ~/2)
j j+k ~
~,.. J
and likewise for
(4.14)
Thus we can restrict our attention to sums over the set
~\R1'
Next, we reduce to the case of the fractional Brownian motion.
Observe
that by (4.8),
. = IJl(A A + ) A2(J,k)
j
j k
> (1-~) ~T
P(g(~)
~(2
j+k ):
gk(~) > (1-~)~Tk ~(2j»1 < C(k+j)215 2-2jHl5.
Thus
n
(4.15)
On
:I Y_
~2
A (j k) ~ C :I 2-2jHl5
2
'
j=1
:I
(k+j)215 = 0(1).
k~j2l~
R2\R1' we can reduce to the case of independent random variables.
By
(4.11),
(4.16)
:I
~
2
\R
1
A1(j,k)
~
n
C:I
j=1
:I
2
-kG
= 0(1).
k~n~/2
And for independent events, (4.12) is obvious.
We note that this proof has not taken (4.9) and (4.10) into account.
can be trivially modified to do so.
It
-30-
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Cambridge University
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Current Address (beginning May 1989)
Department of Mathematics
Indiana University
Bloomington. IN 47405
.