L - Department of Statistics - University of Washington
LIMIT THEOREMS AND INEQUALITIES FOR THE UNIFORM EMPIRICAL PROCESS INDEXED BY INTERVALS
BY
JON A, y/ELLNER GALEN R. SHORACK
TECHNICAL REPORT NO. 2 JUNEJ 1980
RESEARCH SUPPORTED BY THE NATIONAL SCIENCE FOUNDATION UNDER
GRANT MCS 77-02255 AND GRANT MCS 78-09858
DEPARTMENT OF STATISTICS UNIVERSITY OF WASHINGTON SEATTLE} WASHINGTON
i
l"u\J\!l\L
I
VI t
of
1 ner
l'UJUALU
en
and
he ter anJ
Shorac
ers it
-
of has
60ros,
1,
t
ec
l
i 1
rl t
t
+
t
l
s
1
; !
,
L f'.1
l\e
rie r
9
June
1
neh expon roc
are est
tl
ts " an
5
C
i nt c r,
i J 1\' r
s.
l
r t
s t r i ra
Thes
concern:
t
e conve r t
+
the
A stron
15
for
rem re ated to t}lC well-known
t
Bro~nian
rid
~
1
also
Conn
Kie er and Stute are mentioned.
As an
stat
~h
f t
t c fOT testin
~ork
relat
ti
uniforffiit)" T
S +'-
r , E ck r , of LsaK
ication we introduce
en is the natural +
c
1
+
L
t
new
r va l ana I
U
(1. 1
denote
t
proce
t
for
t)-tJ
is
t
~ell-kno~n t
a
n
t
t
d + L
T
functional
that is
useful functi
ful,
ar ~e
.,
[
.5
conti
s
metr
result
j
8..
not d fi tr
1
I
t
distanc
bet~een
ions
f
.,
, c
,I,
1e t
sup
- sup
h
tive and negat
the
t
t
Let h+ and
b
1.3
>
.37
follows
Now
.19
is suf
prove
-
(1-0) M (log n)
is now
a
=
n
-1
By
, it suf f Lce s
23
s Clsb
(I
lJ!
IIII
31,. n
JJ
[0 (log n) a/2
40n 3
(log n )
a
exp
-(I-D)
by (1.19)
s
for
constant· n
M sufficiently large.
t
>
0
1
4
2 M
---2
(log
n)
(log log
2-a n)
2
II
20(1-a) (log log n) M3 (log n)
l-ex
/
(log log
n)
J
24
REFERENCES Bennett, G. (1962).
Probability inequalities for the sum of independent
random variables.
J. Amer. Statist. Assoc. 57 33-45.
Bolthausen, Erwin (1977).
A non-uniform Glivenko-Cantelli theorem.
Preprint.
Cassels, J. W. S. (1951 .
An extension of the law of the iterated logarithm.
Proc. Camb. Phil. Soc. 47 55-64. Chibisov, D. M. (1964).
Some theorems on the limiting behavior of empirical
distribution functions. Probabili
Selected Transl. in Math. Statist. and
6 147 -1 S6 .
An est
concern
the Kolmogorov limit distribut
Trans. Amer. Math. Soc. 67 3b-50. Chung, K. L., Erd~s, P., and Sirao, T. (1959).
On the Lipschitz condition
.
for Brownian motion. J. Math. Soc. Janan 11 263-274 . (saki, E. (197
The law of the iterated logarithm for normalized empirica
distribution function.
Wahrscheinlichkeitstheorie verw. Gebiete 3
147-167. Csorgo, ~l. and Reve s z , P. (1979). process?
Ann. Probabili
Durbin, J. (1973). Soc. Ind.
Regional Conference Series in Applied
~lat
ema t
Math. 9 1-64.
Eicker, F. (1979).
The asymptotic distribution of the suprema of t
standardized
ical processes.
w.
Ann. Statist. 7 116-138.
Probability
vari
ities for sums of
Amer. Statist. Assoc. 5
Ito, K. and McKean, H. Pa
st ----
7 731-737.
Distribution Theory for Tests Based on the Sample
Distribution Function.
Jaeschke,
How big are the increments of a Wiener
13-30.
Diffusion Processes and Their
---
BY
JON A, y/ELLNER GALEN R. SHORACK
TECHNICAL REPORT NO. 2 JUNEJ 1980
RESEARCH SUPPORTED BY THE NATIONAL SCIENCE FOUNDATION UNDER
GRANT MCS 77-02255 AND GRANT MCS 78-09858
DEPARTMENT OF STATISTICS UNIVERSITY OF WASHINGTON SEATTLE} WASHINGTON
i
l"u\J\!l\L
I
VI t
of
1 ner
l'UJUALU
en
and
he ter anJ
Shorac
ers it
-
of has
60ros,
1,
t
ec
l
i 1
rl t
t
+
t
l
s
1
; !
,
L f'.1
l\e
rie r
9
June
1
neh expon roc
are est
tl
ts " an
5
C
i nt c r,
i J 1\' r
s.
l
r t
s t r i ra
Thes
concern:
t
e conve r t
+
the
A stron
15
for
rem re ated to t}lC well-known
t
Bro~nian
rid
~
1
also
Conn
Kie er and Stute are mentioned.
As an
stat
~h
f t
t c fOT testin
~ork
relat
ti
uniforffiit)" T
S +'-
r , E ck r , of LsaK
ication we introduce
en is the natural +
c
1
+
L
t
new
r va l ana I
U
(1. 1
denote
t
proce
t
for
t)-tJ
is
t
~ell-kno~n t
a
n
t
t
d + L
T
functional
that is
useful functi
ful,
ar ~e
.,
[
.5
conti
s
metr
result
j
8..
not d fi tr
1
I
t
distanc
bet~een
ions
f
.,
, c
,I,
1e t
sup
- sup
h
tive and negat
the
t
t
Let h+ and
b
1.3
>
.37
follows
Now
.19
is suf
prove
-
(1-0) M (log n)
is now
a
=
n
-1
By
, it suf f Lce s
23
s Clsb
(I
lJ!
IIII
31,. n
JJ
[0 (log n) a/2
40n 3
(log n )
a
exp
-(I-D)
by (1.19)
s
for
constant· n
M sufficiently large.
t
>
0
1
4
2 M
---2
(log
n)
(log log
2-a n)
2
II
20(1-a) (log log n) M3 (log n)
l-ex
/
(log log
n)
J
24
REFERENCES Bennett, G. (1962).
Probability inequalities for the sum of independent
random variables.
J. Amer. Statist. Assoc. 57 33-45.
Bolthausen, Erwin (1977).
A non-uniform Glivenko-Cantelli theorem.
Preprint.
Cassels, J. W. S. (1951 .
An extension of the law of the iterated logarithm.
Proc. Camb. Phil. Soc. 47 55-64. Chibisov, D. M. (1964).
Some theorems on the limiting behavior of empirical
distribution functions. Probabili
Selected Transl. in Math. Statist. and
6 147 -1 S6 .
An est
concern
the Kolmogorov limit distribut
Trans. Amer. Math. Soc. 67 3b-50. Chung, K. L., Erd~s, P., and Sirao, T. (1959).
On the Lipschitz condition
.
for Brownian motion. J. Math. Soc. Janan 11 263-274 . (saki, E. (197
The law of the iterated logarithm for normalized empirica
distribution function.
Wahrscheinlichkeitstheorie verw. Gebiete 3
147-167. Csorgo, ~l. and Reve s z , P. (1979). process?
Ann. Probabili
Durbin, J. (1973). Soc. Ind.
Regional Conference Series in Applied
~lat
ema t
Math. 9 1-64.
Eicker, F. (1979).
The asymptotic distribution of the suprema of t
standardized
ical processes.
w.
Ann. Statist. 7 116-138.
Probability
vari
ities for sums of
Amer. Statist. Assoc. 5
Ito, K. and McKean, H. Pa
st ----
7 731-737.
Distribution Theory for Tests Based on the Sample
Distribution Function.
Jaeschke,
How big are the increments of a Wiener
13-30.
Diffusion Processes and Their
---
