A Note on the almost everywhere central limit theorem
Michael T. Lacey and Walter Philipp
Department of Statistics. UNC. Chapel Hill
1.
Introduction.
The theorem in the title refers to a recent result in [2]
and [11]. (also see [3]). which is as follows:
Let
Sk
be the k-th partial
sum of i.i.d. real-valued random variables (r.v.s) Xi with mean O. variance
1 and finite (2+6)-th moments (6)O in [2] and 6=1 in [11]).
be defined on a probability space
(O.~.P).
Let the r.v.s
Then there is a P-null set NCO
c
·e
such that for all w € N •
(I)
•
for all Borel sets A
x €
(2)
m.
c
mwith
A{8A) = O.
If 6{x) denotes the point mass at
For all w € Nc
then (I) can be restated in this way.
(log n)
-1
~~n
k
-1
6{k
~
Sk{w»
t::J
~N{O.l).
The functional version of (2). proved in [2]. is as follows.
Define the
usual "broken line process" on [0.1].
(3)
s (t.w) =
n
{
n~~
if
t
= kin.
(log n)
-1
= O.l •...• n
linear in between
and denote by 6{x) the point mass at x € C[O.l].
(4)
k
~~n
-1
k
6{sk{-'w»
~
~
W
Then
P- a.s.
- 2 -
Here W is standard Brownian motion on [0.1].
In this note we prove the following results.
Theorem 1.
(O.~.P).
~
Let {X • j
j
I} be any sequence of real-valued r.v.s defined on
Suppose there exists a sequence {Y • j
j
~
I} of i.i.d. N(O.l) r.v.s
such that with probability 1
(5)
Then (4) holds.
~
Theorem 2.
Let {X • j
variance 1.
Then (4) holds.
j
sequ~nce
I} be a
of LLd.r.v.s with mean 0 and
The difference between the proof of Theorem 1 and that in [2]
two-fold.
is
First. we assume an almost sure invariance principle (5) (ASIP)
and present a simple/.f~~P~~r1Umtlte"theorem to the Gaussian case.
1.<...1
Second. _
_"
..
•.
the latter case is done in a straightforward. probabilistic way. with no
appeal to ergodic
th~~~~qa;:'in [2].
The proof of Theorem 2 is a corollary
oj1n~. f\;'
to
the proof of Theorem 1.
Further remarks on the
theorems and some
extensions are at thf)C!!ng of n~s )~O,~.
2.
Proofs.
~
Denote by BL = BL(C[O.l]. 1I-lIco )
f : C[O.l] -+ IR wi th IIfll
BL
: = IIfll
L
+ IIfll co
<
00.
the class of functions
Here
IIfIlL := sup{lf(x) - f(y)I/llx-ylloo : x.y € C[O.l]. x ~ y}.
We first prove that (4) is equivalent to the following statement:
f € BL
(6)
(log n)
-1
~~n k
-1
f«sk(-'w»
-+Ef(W(-»
P-
a.s.
For each
-
- 3 -
Although the left-hand side of (4) does not define a probability
measure, it is so close to one that the theory of weak convergence still
applies.
But (6) does not yet imply (4) since, in general, the exceptional
P-null set may depend on f.
Theorem 8.3, II
However, it is clear from the proof of [5,
=> III] that if we have (6) for a certain countable subset
of BL, which is not difficult to write down explicitly, then we have only a
countable union N of P-null sets to contend with, and,
convergence in (6) is uniform in f € BL.
~
Next, by (3) and (5),
th~re
outside N,
the
Thus (4) and (6) are eqUivalent.
is a P-null set, call it also N, such that
C
for all 6J € N
If(Sk(e,6J» - f(t (e,6J}) I ~ ,~lfIlBL 1I~(e,6J) - t (e,6J)lI oo
k
k
"
·e
Here we denoted by t
"I.l
....._ .
n
Thus (4) is equivalent to
replaced by Y .
j
'\1-
m~1,
f
P- a.s.
(7)
for each f € BL.
Hence we have reduced the proof to the case of Gaussian
r.v.s.
For the proof of (7) we note that t
£.:.
{Y , j
j
(8)
~
1}, i.e.
E
f(~)
(log n)
-1
-+.E feW).
lc~n k
where
(9)
We need the following estimate.
-1
~
k
-+W by Donsker's theorem applied to
Hence it is enough to prove that
f k -+ 0
P- a.s.
- 6 -
~
The use of the logari thmic mean looks perhaps peculiar at first glance,
e
but it apPears that it is essentially the only sUJllll8.tion method that works.
The Cesaro mean certainly does not, since
~ #{k ~ n; ~ > O} ~ arcsin.
A similar argument shows that the weights k-a, 0
<a <1
don't work either.
Moreover, the covariance estimate (10) is sharp, and the present method will
not allow the use of any sUJllll8.tion method which is not close to the
logarithmic mean.
L.. The rates of convergence ip these theorems cannot
be o(l/log n) as can
be seen for instance by setting A = IR in (1).
e-
REFERENCES
[1]
[2]
J. Wiley,
Billingsley, Patrick,
N.Y .. 1968.
Convergence of probability measures,
Brosamler, Gunnar A.,
An almost everywhere central limit theorem,
Math.. Proc. Qunb. PhH. Soc. 104. 561-574 (1988).
[3]
Brosamler. Gunnar A.• Un theoreme limite presque sur pour les
mesures d'occupation du mouvement brownien sur une variete
riemannienne compacte. C.R. Acad.. SeL 307. Ser. I, 919-922 (1988).
[4]
Chung. K.L.. A course in probability theory.
Academic Press. New York. 1974.
[5]
Dudley. R.M .• Probabilities and Metrics. Lecture Note Series 45.
Aarhus Universitet. 1976.
[6]
Dudley. R.M. and Walter Philipp. Invariance principles for Banach
space valued random elements and empirical processes. Z.
WahrschetnUchkeHsth.eorte venD. Geb. 62. 509-552 (1983).
[7]
Einmahl. Uwe. Strong invariance principles for partial sums of
independent random vectors. Annals Prob. 15. 1419-1440 (1987).
Second edition.
- 7 -
[8]
Philipp. Walter. Invariance principles for sums of mixing random
elements and the multivariate emprirical process. Cott. Math. Soc.
Janos Botyat 36 (Limit Theorems in Probability and Statistics).
843-873. Veszprem (Hungary). 1982.
[9]
Philipp. Walter. Invariance principles for independent and weakly
dependent random variables. In Dependence in Probability and
Statistics. E. Eberlein and M.S. Taqqu. ed. Birkhauser. Boston 1986.
If
[10] Philipp. Walter and William Stout. Almost sure invariance principles
for partial sums of weakly dependent random variables. Memoirs AMS
161 (1975).
[11] Schatte. Peter.
On strong versions of the central limit theorem.
Math. Nachr. 137. 249-256 (l988).
Michael T. Lacey
'i~
Wal ter Philipp
Departments of Mathematics
~,;-;)s~:: '''and Statistics
University of Illinois
Urbana. IL 61801
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