Math. Proc. Camb. Phil. Soc. (1988), 104, 561
Printed in Great Britain
561
An almost everywhere central limit theorem
BY GUNNAR A. BROSAMLER
Fachbereich Mathematik, Universitdt des Saarlandes, Saarbriicken, West Germany
and Department of Mathematics, The University of British Columbia,
Vancouver, B.C., Canada
(Received 24 November 1987; revised 28 January 1988)
1. Introduction and summary
The purpose of this paper is the proof of an almost everywhere version of the
classical central limit theorem (CLT). As is well known, the latter states that for IID
random variables Yx, Y2,... on a probability space (Cl,<H,P) with EYX = 0, EY\ = 1,
we have weak convergence of the distributions of (l/\/w)S"_ 1 Yt to the standard
normal distribution on R. We recall that weak convergence of finite measures fin on
a metric space $ to a finite measure fi on S is defined to mean that
I/M.
for all bounded, continuous real functions on S. Equivalently, one may require the
validity of (l g l) only for bounded, uniformly continuous real functions, or even for
all bounded measurable real functions which are /i-a.e. continuous.
If for xeM we denote by Sx the probability measure on U which assigns
its total mass to x, we note that the distribution of n~*Sn, where Sn = 2"_x Yt, is just
iadP((o)SSn(lil)IVn i.e. the average of the random measures Ss M/y/n with respect
to P.
In this paper we shall form 'time averages' with respect to a logarithmic scale
rather than ' space averages' and prove a.e. convergence for the resulting random
measures. To be precise, we have the following
THEOREM 1-2. Let Yv Y2,... be IID random variables on (Q,2T,P) with EYX = 0,
EY\ = \, E\Yx\i+2S < oo for some S > 0. Let Sn = S j ^ Yk, forn^l.
Then P-a.e.
ivhere N(0,1) is the standard normal distribution on IR, and the convergence is weak
convergence of measures on R.
Rephrasing this theorem, we have outside one single exceptional set of Pprobability 0,
, n
for all bounded measurable functions /:(R->(R, which are continuous a.e., or
^
, .
«v
_
,
,
e, u i va,en l l y
AA
for all Borel sets A^U
\Vkj
V(2n))
with Leb (dA) = 0.
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562
GUNNAR A. BROSAMLER
To compare Theorem 1-2 with the classical CLT one might consider using (13) to
test a random number generator. One would only have to check one (typical)
sequence, rather than many sequences as in tests based on the classical CLT. In [4],
we shall use a version of such an a.e. CLT to recover the spectrum of a compact
Riemannian manifold from the development over time of the energy of one single
diffusion path.
In order to get results like (1-3) for f(SJ\/k,...,8J^k)
rather than just for
f(Sk/\/k),
we shall actually prove a C[0, Inversion of Theorem 1-2, which
corresponds to the Donsker version of the classical CLT and immediately implies
Theorem 1-2. For background material on the Donsker Theorem we refer the reader
to [8], pp. 68-84. Following the notation there, we define for a sequence of real
numbers {sn, n ^ 1} the sequence {s(n),n ^ 1} of elements in C[0,1] by setting s0 = 0,
= 7i~5Sj
for
i = 0 , . . . , n,
and by letting s(n) be linear on [(i — l)/n,i/n] for i = l,...,n. We shall denote by
/iw the Wiener measure on C[0,1], and for xeC[0,1] let 8X be the probability
measure on C[0,1] which assigns its total mass to x. We have
THEOREM 1-4. Under the assumptions of Theorem 1*2, P-a.e.
where the convergence is weak convergence of measures on C[0,1].
This theorem can be rephrased thus: outside a single set of P-measure 0, we
have
for all measurable <p: C[0, l]-> U which are continuous /%-a.e. Here <f>{jiw) is the
image measure of /tw under <f>. Letting in particular ifi^x) = x(l), we obtain Theorem
1-2. Letting 4>%{x) = max U6[0 1]|a;(M)|'x, where a. > 0, we obtain that P-a.e.
1
"
}Z7Z SS kk X*«:-•" max{|S,r
1O
n
6
*-l
*
\Stf) = 02(/*H')-
0 ( )
We may also obtain pointwise asymptotic results for functionals such as the relative
frequency of positive Sk's or the last change of sign in {Sk, k < n), functionals which
will have to be approximated by functions on C[0,1]. To be precise, let So = 0,
•max{0 ^j<n;8j^
0} if Sn > 0
Cn=\ ma5T{0^j<n;8t>0}
.0
if Sn < 0, {} * 0
if S n <0,{} = 0 .
We have the following
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An almost everywhere central limit theorem
COROLLARY
563
1-5. Under the assumptions of Theorem 1*2, P-a.e.
(2)
lin
where AS is the arc sine distribution on [0,1], and the convergence is weak convergence
of measures.
In order to prove Theorem 1-4, we shall first prove a pointwise CLT for onedimensional Brownian motion and then use Skorokhod's representation theorem to
deal with the case of IID random variables.
So let W be one-dimensional Brownian motion starting at 0, on some probability
space (Q, 91,P), and define the C[0, l]-valued random variables W(s), for s > 0, by
By averaging the random point measures SwwM on C[0,1] on a logarithmic time
scale, we get the random measures
1 P
M<*>) = j — t I ds/sSwwM,
(t > 1),
and their discrete analogues
fc-1
We shall prove
THEOREM
1-6.
(a) \im/it = /iw
P-a.e.
(b) lim vn = fiw
P-a.e.
n~*oo
The convergence in Theorem 1/6 is again weak convergence of measures on C[0,1].
Again it follows that outside a single set of P-measure 0, for all measurable
<j>: C[0, l]->-IR which are continuous /iw-&.e.,
1 f
Urn-—- ds/sS,oWw
9
log'J
= ^{/iw),
(1)
lim ^— S i"1 £,.„*> = tM-
(2)
Functions <j> one might want to consider in this context, include the functions
x\-+x{l), max \x(u)\a, Leb{ue[0,1];
x(u) ^ 0},
max{zte[0,1]; x(u) = 0}.
U£[0, 1]
The proof of Theorem 1-6 relies heavily on the process Xs = Ws/y/s, s > 0. This
process is stationary with respect to multiplication on the time scale, which means
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564
GUNNAR A. BBOSAMLER
that for h > 0, the processes {Xs,s>0} and {Xsll,s > 0} have the same laws.
Stationarity allows extensive use of ergodic theory. We remark that the process
{Xes,seM} is the Ornstein—Uhlenbeck process.
In a last section we shall discuss the relationship between our results and results
obtained recently by A. Fisher.
2. The Brownian motion case
We shall first prove that the measures {/it, t^e} of Theorem 16 (a) are tight, P-a.e.
Denning <j>: C[0,1] -*• U by <j>(x) = x(0), we see that the /^-distribution of 0 is So for all
t > 1, P-a.e., since W^ = 0 for all s > 0, P-a.e. Therefore, in order to prove tightness
of the family {/it,t ^ e}, P-a.e., it is, by theorem 12-3 in [1], sufficient to show that
P-a.e., there exists a finite C(w) such that for all ttj,«2e[0,1]
\x(u1)-x(u2)\4/il(co)(dx)^C(oj)\u1-u2\^,
sup
t^e
JC[0,1]
or equivalently,
s u p - i - f ds/sWV^JVs-W^JVM*
< CMK-ua|l.
(2-1)
We shall start with the following Lemma 2-2, which we shall prove, for the reader's
convenience. This lemma follows also from the general theory of Gaussian processes.
LEMMA
2-2.
sup
\Wu+h-Wu\/VWogh\)eL>>,
foraUp>l.
«6[0,
1]
0<h<e~ 2
Proof. Let
^
z =
\Wu+h-WJ/VWogh\).
ue[0,l]
We have for A ^ 0,
{
fc-2
sup
I
f
fc-2
sup
I
sup
\Wv+w-Wv\>Ae-'W(lc+i)\
*-2
£
P{
sup
fc-2 (-0
S (e*+l)pf sup
fc-2
l
S (e* + l)p{2 sup
fc-2
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An almost everywhere central limit theorem
565
*-2
co
(•
-I
t-2
I
J
Letting Ao = \/(2/c2), we have for A > Ao
It follows that for A ^ Ao
P{Z ;* A} s: c, £ exp{-|c2A2A}
t-2
roo
exp {—
This implies J ^ A " - 1 / ^ ^= A}rfA < oo for all p ^ 1.
Remark 2 3 . The preceding lemma implies immediately that for all e < \ and all
P>1
'
sup
K-^nH^-H
The inequality (2-1), hence tightness of the family {jit,t ^ e}, P-a.e., now follows
immediately from the following lemma.
LEMMA 2-4. Por all p ^ 1,
sup
sup
- L — - L - 3
P
i
2
Proof. The random variable in the lemma is majorized by
1
where
Moreover,
1 f*
1 p'
Z* = sup
ds/sZg = sup ds/sZs,
l>e l ° g U l
(51 ' Jl
ffu ,-Wu ,|«
/7
!
Z s = sup
4—
Z, =
sup
(ui—u1)~^\\/
where X 8 = WJy/s for s > 0. As {Xt,s > 0} and { I , 4 , s > 0} have the same laws
for h > 0, {Zs, s > 0} and {ZsA, s > 0} have the same laws for h > 0. By Remark 2-3,
Zj e Lv for p ^ 1. I t follows by the dominated ergodic theorem that
t
re-
supn^l
n
j »-i f*+i
ds/sZ, = s u p - S
Jl
n^l
n
duZeueLp,
k-0Jk
for p > 1. Now we have for t ^ 1,
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566
GUNNAR A. BROSAMLER
1 C e"
Z*^2supds/sZs,
which implies
hence Z*eLp for p 5* 1.
Proof of Theorem 16(a). Let ke N, 0 < ux <... < uk ^ 1, acv ..., <xke R. Let
<p(x) = e*K*(«i>+-••+«**<«*>].
(2-5)
We want to show that P-a.e.
lim I
<j>(x)til{o>){dx) = I
«^oo JC[O,1]
<f>(x)/iw(dx).
(2-6)
JC[O,1]
Letting
Y. = exp(i[a 1 V«i^« 1 . + .
'
^
one can easily check that
f
c[o,ij
Since the laws of {Xs, s > 0} and {Xsh, s > 0} coincide for A > 0, it follows that the laws
of {*Fg,s > 0} and {^A.S > 0} coincide for h > 0. The ergodic theorem and the fact
that Brownian motion has a trivial tail-field imply that P-a.e.
1 f"
lim -
n-»oo n J l
Moreover
1 p'
_[*] J_ pm
1 p11
_ 1 | p'
1 P
P'
lim -- ds/sWs = EWV
so that P-a.e.
t-.00 ' J l
Obviously Ex¥1 = J c[ o,i]0(^)/*H'(^)- This proves (2-6). I t follows that, P-a.e., (26) is
valid for all <f>, defined by (25), if k runs through N, 0 < w2 < ... < uk < 1 run
through the rationals in [0,1] and <x1,...,ak run through the rationals in IR. In view
of the tightness of the family {jit,t ^ e}, proved earlier, this completes the proof of
Theorem 1-6 (a).
We now turn to the proof of Theorem 1-6 (6) and define the C[0, l]-valued process
{Z..s>l} by
Z
for k
LEMMA
^
s
<
k
+
l
(2
.7)
2-8. We have P-a.e.,
lim ||Zg-W<*> ||C[Oj„ = ().
(2-9)
Proof. We have to prove that P-a.e.
lim sup
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An almost everywhere central limit theorem
567
Now
lc[O,l] — S U P
•s/n
U£[O,1]
\/(n
1
^ — - sup \Wu{n+n)-Wun\+
V"ue[O,l]
^TTT
|
SU
sup |
ue[O,l]
P l ^ n + W - ^ - n l + o I ! SUP
I t remains to show that P-a.e.
l i m - J - sup sup |W u ( n + A ) -WU = 0
lim — sup sup \Wu(n+h)\ = 0.
and
(2-10)
(2-11)
n->oo ^ s ue[0,l]
But (2-10) follows from the Borel-Cantelli Lemma and the estimate
P { sup sup
\Wutn+H)-Wun\>erA
lue[0,l] AG[0, 1]
i
^ P { sup sup \WV+S-Wv\>en^
n-l
f
-J
^ S P | sup sup |W0+4—H^J > e»*[
t-0
Ue[I,(+l] 46(0,1]
up
J
s u p | W U + , - W J ^ e»*>
e[0,l] 46[0,l]
2 sup \WV\
U6[0,2]
< en
As for (211), we use the estimate
sup
sup \WMn+h)\ ^ sup
UE[0,1] Ae[0,l]
Moreover, P-a.e., for n ^ e2
1
su
p '^'<rop ^:loglogs)<°°-
This implies (211).
The following lemma is the analogue of lemma 1129 in [8].
212. (a) / / X,Z: R + ->C[0,1] are measurable and are such that
X,—Z,[| ct0 XJ = 0, then for all bounded, uniformly continuous functions
LEMMA
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568
GTJNNAK A. BROSAMLER
(b) IfX,Z: N-^C\Q,\.} are measurable and are such that \\mn^a>\\Xn — Zn\\CV0l] = 0,
then for all bounded, uniformly continuous functions j': C[0,1]->IR,
lim {-L. s k-yf{Xlc)-J— £ k-lf(Zk)\ = 0.
The proof is simple and left to the reader.
Proof of Theorem 16 (6). According to Theorem 16 (a), we have P-a.e.
lim r-t— Tds/sflWM) =(
»-*oo lo g W Jl
f(x)dfiw(x),
(2-13)
Jc[0,l]
for all bounded, continuous/: C[0,1] -> IR. From Lemmas 2-8 and 212 (a) we conclude
for the process {Zs,s ^ 1}, defined by (2-7), that outside the two exceptional sets for
(2-9) and (213), both of P-measure 0,
lim—— ("ds/sf(Zs)=
(
n^ooJog^Jl
Jc[0,l]
f(x)daw(x),
(2-14)
for all bounded and uniformly continuous functions / : C[0,1]-*- U, hence for all
bounded and continuous functions/: C[0, l]-> IR. This however implies that P-a.e.
= f
lim
n->oo
l o
g W fc-l
f(x)d(iw(x)
(2-15)
for all bounded and continuous functions/: C[0, l]-> IR, since
i:
n
n-l
ds/sf(Zs) = S
3. TAe //Z) variables case
We recall a theorem of Marcinkiewicz-Zygmund, as formulated e.g. as theorem 2 in
[5], §52. This theorem states that for IID random variables XltX2,... on some
probability space (£2, 21,P) for which E\X1\1+' < oo for some £e(0,1), we have P-a.e.
lim ( J Xt-nEXAjnl-P = 0,
(31)
^"TT^'
(3 2)
where
'
-
Equation (3 l) implies in particular that P-a.e.
sup
j—nEX j
nl~t> < oo.
(3-3)
Remark. Under the above assumptions, one can also prove that for all
pe (0,8/1+8) there exists C such that for A > 0
Pjsup
U
l
n
-P
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An almost everywhere central limit theorem
569
The proof is based on the estimate
1+S
^CnE\X1-EX1
E
\1+S
The corresponding result for certain mixing random variables is used in [4].
Assume now that on (Q, 51, P) there are defined a one-dimensional Brownian
motion W and random variables 0 = T0 < TX < T2 < ... such that
(34) (1), Tt —Tt_!, k ^ 1, are IID random variables;
(3-4) (2), Erx = \;
(3-4) (3), ET\+S < oo for some Se (0,1).
We shall prove
LEMMA
35.
lim \WTk-Wt\/Vk
= 0, P-a.e.
Proof. Let p be as in (3-2). I t follows from (33) that, for all e > 0, we may choose
A > 0 such that
As e > 0 is arbitrary, it suffices to prove that P-a.e.
\\m\WTn-Wn\/y/n = 0,
n-*oo
on the set
Q,, = |sup \rk — kl/k1'1" < A | .
Now, for 8 > 0 and n e N, we have
def
P(
sup
Distinguishing the cases n < s and 5 <ra,we see that
sup
1
To be more precise, in the first case we use the fact that {Wt,s ^ 0} and {W,+n — Wn,
5 ^ 0 } have the same laws, and in the second case we use the fact that {Wg, 0 ^ s ^ n}
and {Wn_s — Wn,0 ^s ^n} have the same laws.
As {\/(An1~p) Wt,t > 0} and {WAni-rt,t ^ 0} have the same laws, it follows that
an(A) ^ 2P j sup \Wt\ ^ 5A^7^/2] < 8P{WX
It follows that S^_xan(A) < oo, and the proof is finished by the Borel-Cantelli
Lemma.
Proof of Theorem 14. By a theorem of Skorokhod (see e.g. [8]), there exists a
probability space which supports a Brownian motion W and random variables
0 = T0 < Tj < T2, ..., satisfying (34) (1) and (34) (2), such that the sequence
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570
G U N N A K A.
BROSAMLER
{WT ,WT ,...} has the same law as the sequence {St, S2,...}. Here Sk = Sf_j Y{. We shall
henceforth identify S1,S2,... and WT, WT,
Moreover T, can be constructed as the
first exit time of W from a bounded random interval, independent of W. (See [8].)
Applying a Burkholder-Gundy inequality (see e.g. [6], theorem VII, 92) and the
maximal inequality to the martingale WtAT, we have
Sit
For the special construction of TX mentioned above,
E\WIATI\2+2S
< E\WTJ\2+2S
=
which is easily verified if T1 is the first exit time of W from a fixed bounded interval.
Thus
1I
Letting <->• oo, we conclude that T1 satisfies (34) (3). Hence Lemma 35 is valid for W
and Tk of the Skorokhod representation above. Let W(n),S(n) be as in §1, and define
Zn: O -> C[0,1] by setting Zn(i/n) = WJV™ for * = 0,..., n, and letting Zn be linear
on each interval [(i— l)/n, i/n], i = \,...,n. In view of Theorem 1-6(6) and Lemma
2-12(6), it is sufficient to prove that P-a.e.
lim | W(n)-S(B) II CIo, !] = <>.
(3-6)
n-»oo
In order to prove (3-6), it is sufficient to show that P-a.e.
lim | >F<B>-Z" 1 1 ^
=0
(3-7)
lim||Z n -<S (n) || C[0il] = 0.
(3-8)
n-»oo
and
n-»oo
Now (37) follows by a minor modification of the proof of (1131) in [8]: for all
e>0,
P{\\W<n)-Zn\\C[0A] > e} ^ V P { sup \Wt-Wk
fc-0
sup \Wt\ $= \eVn] ^ 4nP{W1
J
By the Borel-Cantelli Lemma we have (3-7).
Equation (38) follows from (35). Indeed,
and the right side converges to 0 if and only if (i/-\/k)\Wr —Wk\ converges to 0.
Proof of Corollary 1-5. Define <f>3, &4: C[0, l]-> R by
<}>3(x) = Lebesgue measure {ue[0,1]; x(u) > 0}
A
9 (
* '
fsup{tt6[0,l];x(«)^0}
isup{ M e[0,l];x(M)>0}
if z ( l ) > 0
if x{l) ^ 0.
Here sup{} = 0, if {} = 0.
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An almost everywhere central limit theorem
571
The functionals <f>3,04 are /iw-a,.e. continuous. (See e.g. pp. 78-79 in [8].) It follows
from Theorem 14 that P-a.e.
lim - i - S k-i S^oS
M
= 4tfrw)
(i = 3,4).
Now P-a.e.,
lim|0 3 o<S ( n ) -LJ = 0,
liml^oS^-n^CJ = 0
n-*oo
n->oo
and
(f>3(jiw) = <f>4(/iw) = AS
(see pp. 78-79 in [8]). The proof is completed by applying the analogue of Lemma
2-12(6) for X,Z: N^U.
Remark 3-9. According to a theorem of Prohorov, a family of probability measures
on a separable complete metric space is relatively compact if and only if it is tight
(cf. theorems 61 and 62 in [1]). It follows that not only the family {/it,t > e} in
Theorem l-6(a) is tight F-a.e., but also the family {vn,n > 2} in Theorem 16(6), as
well as the families of random measures in Theorems 1-2 and 1-4.
Remark 310. It is possible to give a direct proof of Theorem 12, avoiding the
tightness argument which appears to be indispensable in the proof of Theorem 1-4.
In Theorem 1-2 we are considering finite measures /in, fi on S = U, and in this case
fin converges weakly to /i if and only if fin(U) -+/i(U) and (1-1) holds for all continuous
/ : S -> U with compact support. (This equivalence is not valid for finite measures /in,
/i on S = C[0,1], where any continuous / : S -*• U with compact support vanishes
identically.) Thus, for the proof of Theorem 1-2, it suffices to prove that (13) holds
outside a set of probability 0, for all / in the class of continuous real functions on R
with compact support. This class, endowed with the sup-norm, is separable. It
therefore suffices to prove that, for a fixed function / : R->IR which is uniformly
continuous and bounded, (13) holds P-a.e. Now by the ergodic theorem we have
lim I/TO | dsf(Wj/y/{e'))
and hence
lim
= Ef(Wx)
(P-a.e.)
ds/sf(WJVs) = EHW,) (P-a.e.).
gU
We then conclude by Lemmas 2-8 and 35 that the left side can be replaced by the
leftside of (13).
lo
Remark 311. It is possible to prove the validity of (13) P-a.e. for certain
unbounded continuous functions/: U -*• U, such as/(x) = \x\2p, where p > 0. The case
< oo for some
p= 1 gives for IID random variables Y1,Y2,... such that E^^21
8 > 0 the variance formula
lim —!— S kr^t-kEYJ'/k
= Var 7X
(P-a.e.).
(3-12)
It appears to be an interesting problem to determine the rate of convergence in the
a.e. results which we have proved.
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572
GUNNAB A. BROSAMLER
4. Comments on'previousa.e. CLT results
Weak versions of a.e. CLT's, at least for one-dimensional Brownian motion, are
not new. For example, in [3], p. 314, 1. 7, we concluded from the stationarity of the
Ornstein-Uhlenbeck process {We>/\/(es),seR} and the ergodic theorem, that for
e > 0 and ceU, P-a.e.
l im _ J _ P
Leb{s€(0,p); WJy/a > c} = — 1 — [ e'^ du,
r-*0 l o g r J r
which is equivalent (by integration by parts) to
(41)
This law is an a.e. CLT for W at 0.
Stationarity of {We>/-\/(e*),seR} as well as logarithmic (and log2-) averaging had
been used previously in [2] to obtain a.e. results which are not available without
averaging.
In [7] Fisher gives two a.e. CLT results for one-dimensional Brownian motion,
which we shall now discuss.
(a) Fisher's first CLT result
This result ([7], p. 215,11. 4-5) is in terms of averages. It states (without proof) that
for/eL°°(IR), P-a.e. the so-called order two (and higher) averaging methods, applied
to the function Fm: (R+-> U, st-*f(Ws{w)/y/s), converge to N(0,1)(/). As
1 P
ds/sF(s)
is a particular averaging method of order two (see below), this includes the following
result: for/e2/°(R), P-a.e.
Urn - i - f'ds/sfiWJVs) = N(0,1)(/).
t->a>
l o
(4-2)
6 f Jl
This latter result is the dual of (41') (if / = #(c „,), and like (4 1') it follows
immediately from the stationarity of the process {We>/-\/(es),seU} and the ergodic
theorem.
We shall show that Fisher's apparently more general result on general order two
(and higher) averaging methods can be deduced from (4-2). The non-probabilistic
Theorem 4-3 below will show that, for any FeLm{U) for which
1 P
lim
ds/sF(s) = c,
l°g<Ji
we have convergence to c, of all order two (and higher) averaging methods. Thus the
general result on p. 215, 11.4-5 in [7] follows from (42) in a deterministic way.
In order to formulate Theorem 43 we recall the definition of the averaging
operators used in [7]. Following [7], we let
1
(R);4)^0, |<Dds= ll.
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An almost everywhere central limit theorem
l
a3
x
573
s
The operators E,E~ : L (R)^L (R)
are defined by (EF){s) = F(e ) for seU,
(E-*F)(s) = F{logs) for s > 0 and = 0 for 5 ^ 0 . For O G P ^ R ) , the operators
AZ:Lco(U)^La>(n),
n^O, are defined by A%F = <1>*F, A^+1 = E~1A^E. The
operator A% is called an averaging operator of order n. For n ^ 1 and Fe L°°(IR), the
function A%F depends only on the values of F on R+. So A%F makes sense for
n ^ 1 and FeL°°(IR+). Letting <to(s) = e~";\;(0>00)(-s), we have for sufficiently large t,
(AlaF)(t) = i1 [tF(s)ds,
Jo
{AltF)(t) = lo- Ll Prfs/SjF(s), etc.
S Ji
In the following theorem " : L1(R)-+LCO(R) denotes the Fourier transform
2nia8
®(a)= \®(s)e
\
ds
(aeU).
4-3. Let FeL°°(U+), ceU. If l i m ^ (AJF)(t) = cforsome neN and some
R) with 6(a) =t= 0 for all a e R , then l i m ^ , (^4J+*J^)(<) = c for all integers k^O
and all WeP^R).
THEOREM
Remark. For cl>o defined above we have 4>0(a) = (1 — 2nia.)~1 for a e R , so that
<I>0(a) + 0 for all a e R .
Theorem 4-3 is probably well known, but for the reader's convenience we recall
that for k = 0, the assertion follows from Wiener's Tauberian Theorem. (For this
theorem see e.g. p. 10 in [9], and keep in mind that we use a slightly different
definition of the Fourier transform.) For k > 1, it follows from the following
4-4. Let FeLx(R) and ceR. i / l i m ^ , (<&*F)(t) = cfor some <DePt(lR), then
l i m , ^ («F* (EF))(t) = cfor all
LEMMA
Proof. As limt_0O(<I>*.F)(0 = c, we conclude that limt^ro (A^i® *F))(t) = c. Now
l i m ^ [A]y((<t>*F)-F)](t) = 0 (see p. 221 of [7]), and therefore l i m ^ {A\.F)(t) = c.
(6) Fisher's second CUT result
This result, theorem 4 1 in [7], is stated in terms of means on L°°(IR).
We recall that a mean on LCO(IR) is a positive, normalized, (continuous) linear
functional A: L°°(IR)->- R. The mean A is called weighted at oo if A(/) only depends on
the values of/ arbitrarily far on the right. In particular, A is weighted at oo if it is
exponential-invariant. Fisher's paper is to a large extent concerned with the
existence of means on L°°(IR) with certain properties. In particular he proves the
existence of a mean A on Lm(R) which is convolution-invariant and exponentialinvariant.
Theorem 4 1 of [7] may be stated as follows:
Let A be a mean on L°°(IR). If (a) A is convolution-invariant and exponentialinvariant or if (6) A is weighted at oo and ' measure-linear', then for any Lebesgue set
A^U, F-a.e.
= N(0, 1){XA).
(4-5)
Under assumption (a) this result is weaker than the averaging result (4*2), in the
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574
G U N N A E A. BBOSAMLER
sense that it can be deduced deterministically from (4-2), but not vice versa. Indeed,
if A satisfies (a), we have A(F) = MA%aF) for FeL<°(U+). Recalling that
for t > 1, we conclude by the remark on p. 219 in [7] (with c(x) = ef) that
If
lim-
i
ds/sF(s)^\(F)^
lim-
ft
ds/sF(s),
(4-6)
which together with (4-2) implies the result.
The result (45) under assumption (b) has no apparent relationship with the
averaging result (42). It may be proved as follows. If A is weighted at oo, the
triviality of the tail-field of W implies that A(th~*XA^t/Vi)) is constant P-a.e.,
and the definition of 'measure-linearity' allows the direct verification that
EMt^xA(Wt/Vt}) = N(0, 1)(XA).
Notice that in [7] the exceptional w-sets depend on/, A. Thus the results in [7] are
not universal results for large classes of / or A. In this respect the formulation of
theorem 4-1 in [7] is perhaps somewhat misleading, as the proof there clearly
indicates that only one fixed A is considered. Together with our Theorem l-6(a)
however, (4-6) implies in a deterministic way that, outside one single set of
probability 0, (45) holds simultaneously for all A satisfying (a) and for all Borel sets
4 c R with Leb (3.4) = 0.
REFERENCES
[1] P. BILLINOSLEY. Convergence of Probability Measures (Wiley, 1968).
[2] G. A. BROSAMLER. The asymptotic behaviour of certain additive functionals of Brownian
motion. Invent. Math. 20 (1973), 87-96.
[3] G. A. BROSAMLER. Brownian occupation measures on compact manifolds. In Seminar on
Stochastic Processes 1985 (Birkhauser-Verlag, 1986), pp. 290-322.
[4] G. A. BROSAMLER. Energy of diffusion paths and the spectrum of the Laplacian (to appear).
[5] Y. CHOW and H. TEICHER. Probability Theory (Springer-Verlag, 1978).
[6] C. DELLACHERIE and P. A. MEYER. Probabilities and Potential B (North-Holland Publ. Co.,
1982).
[7] A. FISHER. Convex-invariant means and a pathwise central limit theorem. Adv. in Math. 63
(1987), 213-246.
[8] D. FREEDMAN. Brownian Motion and Diffusion (Holden-Day, 1971).
[9] H. REITER. Classical Harmonic Analysis and Locally Compact Groups (Oxford University
Press, 1968).
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