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The law of the iterated logarithm for Lp-norms of empirical processes
Statistics
& Probability
Letters
28 (1996)
107- 1 IO
The law of the iterated logarithm for &-norms empirical processes L. Gajek, M. Kahszka”, Institute of‘ Mathematics, Technical University of&id& Received
March
1995; revised
of
A. Lenic ul. Zwirki 36, 90-924 Lode, Poland May
1995
Abstract Let Xi,&, . . be a sequence of i.i.d. random variables with a common continuous distribution It is shown that
where F, denotes the empirical
distribution
Ah4.Sclassification:
60F25, 62630
Keywords: Empirical
distribution
function of the sample Xl,&,
function; Law of iterated logarithm
(LIL);
function F and let pal.
,X,.
L,-limit
theorems
1. Introduction and main result Let F,, denote the empirical d.f. of a sequenceX*,.X,, . .,X, of i.i.d. random variables with a common continuous distribution function F. Smirnov ( 1944) and, independently, Chung ( 1949) proved the following law of iterated logarithm liE:p
( 210~logn)
‘I2 s;p (F,(t) -F(t),
= 2-l,
a.s.
The &-version of the law of iterated logarithm was obtained by Finkelstein (1971):
* Corresponding
author.
0167-7152/961$12.00 @ 1996 Elsevier SSDI 0167-7152(95)00099-2
Science
B.V. All rights reserved
108
L. Gqjrk
Table Some
1 values
CI ul. I Srutistics
& Prohubilit,v
Ldrers
28 (1996)
107- II0
of C(p)
P
1.1
C(P)
0.292
1.2 0.295
1.3 0.299
1.4 0.302
In Lenic (1994) the following
1.5 0.304
1.6 0.308
I.7 0.311
I.8
1.9
3
0.314
0.316
0.339
4 0.356
5 0.371
result can be found
The aim of this paper is to provide a general form of the law of iterated logarithm when the distance between F, and F is measured with respect to the L,-metric, 1 < p
( 210~logn)1”’
[lz
Jimction
,Fn(t) -F(t),“dF(r)]“P
on the real line, and 1 < p < 30. Then = C(P),
a.s.7
where qp)
= ; (P(P,2’)”
(&)‘.P
w;9
Remarks 1. Of course, C(1) = l/(2&), C(2) = n-‘, and lim,,, C(p) = i. More values of C(p) are given in Table 1. A review of results related to the law of iterated logarithm can be found in C&go and Revesz (1981). A survey of results on L,-norms of empirical processes is given in Csiirgii and Horvkh (1993). 2. Proof of Theorem 1 Let .Y[a,t, be the Finkelstein set, i.e. the set of all absolutely continuous functions J‘: [0, l] + Iw such that f(0) = ,f( 1) = 0 and JAf’(t)’ dt < 1. Then the sequence
My)=(210;logn)’ *(F, (invF(y))
- y),
06~~
1,
is relatively compact with probability 1, and ~-[e.rl is the set of its limit points with respect to the sup norm (see Finkelstein, 1971 or Csijrgii and Rev&z, 198 1, Chap. 5). Hence IF,,(t) - F(t)/’
dF(t) = lim sup 11 ,,-30
IBn(F(f))lP
dF(t)
(1) Since SLO,JI is relatively compact in C[O, I] metrized by supremum norm, there is a function, say x(t), for which the supremum in (1) is attained. Without loss of generality, we assume x(t) 2 0, for all 0
L. Gajek
et al. I Statistics
& Probability
Letters
28 (1YYhj
107-110
109
&x’(t)’ dt = 1. F.irst we shall show that x E C’(O, 1). The standard calculus of variation method shows that it is necessary for x(t) to satisfy I I XP-‘(t)y(t)dt + 2R ?(l).y’(!)dt = 0 (2) P J0 s0 for every y E F[a,il (where i is a Lagrange multiplier). (1963) that x E C’(O, 1) and 2ix”(t)
= p+(t),
Multiplying
It follows
from Lemma 2 of Gelfand and Fomin
O
(3)
both sides of (3) by x’(t)
#qX’(t)2 - x’(o)2] = xP(t),
and integrating, we obtain
O
1.
Now, observe that from (3) and the conditions s(O) = x( 1) = 0 and x(t)>O, it follows a concave function on [O,l]. Moreover, by (4) we have x’(l) = -x’(O). Put a = x’(0) Integrating both sides of (4) from 0 to 1 gives
(4) that i. < 0 and x is and b = JA.*p(t)dt.
(5) Since x E C2(0, l), x(t) 3 0 for t E [0, l] and x(O) = x( 1) = 0, there exist two (possibly equal) points ti, t2 E (0, 1) such that x is strictly increasing on [O,ti] and x is strictly decreasing on [tl, 11. From (4) we get JxP(t)/A x’(t)
=
+ x’(O)2
0
L -
xp(t)/j.
+x’(O)*
for f E [O,ti], for t E [tl,t21, for t E [t2, 11.
(6) (6’)
Let us observe that the function on the right-hand side of (6) (and (6’)) is Lipschitzian, as a function of X, on the set [0, tl - E] (and [fl + F,I], respectively) for every sufficiently small E > 0. Hence for every E > 0 there exists a unique solution to (6) (and (6’)) on the interval [0, tl - E] ([tz + E,i], respectively). Since the supremumon the right-hand side of (1) is attained also for the function x*(t) = x( 1 - t), t E [0, 11, the above implies that x(t) = x(1 - t) on both intervals [0, tl) and (t2, 11. This property extends to the whole interval [0, l] becausex is constant on [tl, t2] and continuous on [0, I]. Hence maxagtGi x(t) = x(i). By (4), (5) and the equality x’(i) = 0, we get 2
x(i)
=
I/p
( > -&
.
(7)
It also follows from (4) that
.X(f) J 1
0
&qT$
ds=t
for all Odt
and
From (5), (7) and (9), after simple algebra we obtain
b= 2-P(a2 - I)&-2(6’&,,),.
(8)
110
L. Gajek
et al. I Statistics
& Probability
Letters
28 (1996)
107~110
On the other hand, the definition of b combined with (5), (7) and (8) yields l/2
b=2
I0
xP(t)dt
= 2
(11) Eliminating the constant a from (10) and (1 1), we find that b = (2e(~))-~(S(p,lQ(~))(l where S(P) =
I.I tp o m
Making the substitution S(P) = (llP)K(l where B(s,z)
- S(P>/Q(P))-~:‘~>
dt,
and
Q(P) =
Is
1 ___ 0 Jr-rp
(12)
dt.
u = 1 - tJ’, we get + PYPY ;h
Q(P) = (llP)RlPT
;I>
is the beta function. Finally, from the last two formulas and (12), we obtain that
which completes the proof.
0
References Chung, K.L. (1949), An estimate concerning the Kolmogorov limit distribution, Trans. Amer. Math. Sot. 67, 36-50. C&go, M. and Rtvdsz, P. (1981), Strong Approximations in Probability and Statistics (Academic Press, New York). Csorgii, M. and L. Horvath (1993) Weighted Approximations in Probability and Statistics (Wiley, Chichester). Finkelstein, H. (1971), The law of the iterated logarithm for empirical distributions, Ann. Math. Statist. 42, 607-615. Gelfand, I.M. and S.V. Fomin (1963), Calculus of’ Variations, Revised English ed. (Prentice-Hall, Englewood Cliffs, NJ). Lenic, A. (1994), Projection methods for nonparametric density estimation, Ph.D. Thesis (in Polish), Polish Academy of Sciences. Smimov, N.V. (1944), An approximation to the distribution laws of random quantities determined by empirical data, Uspehi Mat. Nauk. 10. 179-206.
& Probability
Letters
28 (1996)
107- 1 IO
The law of the iterated logarithm for &-norms empirical processes L. Gajek, M. Kahszka”, Institute of‘ Mathematics, Technical University of&id& Received
March
1995; revised
of
A. Lenic ul. Zwirki 36, 90-924 Lode, Poland May
1995
Abstract Let Xi,&, . . be a sequence of i.i.d. random variables with a common continuous distribution It is shown that
where F, denotes the empirical
distribution
Ah4.Sclassification:
60F25, 62630
Keywords: Empirical
distribution
function of the sample Xl,&,
function; Law of iterated logarithm
(LIL);
function F and let pal.
,X,.
L,-limit
theorems
1. Introduction and main result Let F,, denote the empirical d.f. of a sequenceX*,.X,, . .,X, of i.i.d. random variables with a common continuous distribution function F. Smirnov ( 1944) and, independently, Chung ( 1949) proved the following law of iterated logarithm liE:p
( 210~logn)
‘I2 s;p (F,(t) -F(t),
= 2-l,
a.s.
The &-version of the law of iterated logarithm was obtained by Finkelstein (1971):
* Corresponding
author.
0167-7152/961$12.00 @ 1996 Elsevier SSDI 0167-7152(95)00099-2
Science
B.V. All rights reserved
108
L. Gqjrk
Table Some
1 values
CI ul. I Srutistics
& Prohubilit,v
Ldrers
28 (1996)
107- II0
of C(p)
P
1.1
C(P)
0.292
1.2 0.295
1.3 0.299
1.4 0.302
In Lenic (1994) the following
1.5 0.304
1.6 0.308
I.7 0.311
I.8
1.9
3
0.314
0.316
0.339
4 0.356
5 0.371
result can be found
The aim of this paper is to provide a general form of the law of iterated logarithm when the distance between F, and F is measured with respect to the L,-metric, 1 < p
( 210~logn)1”’
[lz
Jimction
,Fn(t) -F(t),“dF(r)]“P
on the real line, and 1 < p < 30. Then = C(P),
a.s.7
where qp)
= ; (P(P,2’)”
(&)‘.P
w;9
Remarks 1. Of course, C(1) = l/(2&), C(2) = n-‘, and lim,,, C(p) = i. More values of C(p) are given in Table 1. A review of results related to the law of iterated logarithm can be found in C&go and Revesz (1981). A survey of results on L,-norms of empirical processes is given in Csiirgii and Horvkh (1993). 2. Proof of Theorem 1 Let .Y[a,t, be the Finkelstein set, i.e. the set of all absolutely continuous functions J‘: [0, l] + Iw such that f(0) = ,f( 1) = 0 and JAf’(t)’ dt < 1. Then the sequence
My)=(210;logn)’ *(F, (invF(y))
- y),
06~~
1,
is relatively compact with probability 1, and ~-[e.rl is the set of its limit points with respect to the sup norm (see Finkelstein, 1971 or Csijrgii and Rev&z, 198 1, Chap. 5). Hence IF,,(t) - F(t)/’
dF(t) = lim sup 11 ,,-30
IBn(F(f))lP
dF(t)
(1) Since SLO,JI is relatively compact in C[O, I] metrized by supremum norm, there is a function, say x(t), for which the supremum in (1) is attained. Without loss of generality, we assume x(t) 2 0, for all 0
L. Gajek
et al. I Statistics
& Probability
Letters
28 (1YYhj
107-110
109
&x’(t)’ dt = 1. F.irst we shall show that x E C’(O, 1). The standard calculus of variation method shows that it is necessary for x(t) to satisfy I I XP-‘(t)y(t)dt + 2R ?(l).y’(!)dt = 0 (2) P J0 s0 for every y E F[a,il (where i is a Lagrange multiplier). (1963) that x E C’(O, 1) and 2ix”(t)
= p+(t),
Multiplying
It follows
from Lemma 2 of Gelfand and Fomin
O
(3)
both sides of (3) by x’(t)
#qX’(t)2 - x’(o)2] = xP(t),
and integrating, we obtain
O
1.
Now, observe that from (3) and the conditions s(O) = x( 1) = 0 and x(t)>O, it follows a concave function on [O,l]. Moreover, by (4) we have x’(l) = -x’(O). Put a = x’(0) Integrating both sides of (4) from 0 to 1 gives
(4) that i. < 0 and x is and b = JA.*p(t)dt.
(5) Since x E C2(0, l), x(t) 3 0 for t E [0, l] and x(O) = x( 1) = 0, there exist two (possibly equal) points ti, t2 E (0, 1) such that x is strictly increasing on [O,ti] and x is strictly decreasing on [tl, 11. From (4) we get JxP(t)/A x’(t)
=
+ x’(O)2
0
L -
xp(t)/j.
+x’(O)*
for f E [O,ti], for t E [tl,t21, for t E [t2, 11.
(6) (6’)
Let us observe that the function on the right-hand side of (6) (and (6’)) is Lipschitzian, as a function of X, on the set [0, tl - E] (and [fl + F,I], respectively) for every sufficiently small E > 0. Hence for every E > 0 there exists a unique solution to (6) (and (6’)) on the interval [0, tl - E] ([tz + E,i], respectively). Since the supremumon the right-hand side of (1) is attained also for the function x*(t) = x( 1 - t), t E [0, 11, the above implies that x(t) = x(1 - t) on both intervals [0, tl) and (t2, 11. This property extends to the whole interval [0, l] becausex is constant on [tl, t2] and continuous on [0, I]. Hence maxagtGi x(t) = x(i). By (4), (5) and the equality x’(i) = 0, we get 2
x(i)
=
I/p
( > -&
.
(7)
It also follows from (4) that
.X(f) J 1
0
&qT$
ds=t
for all Odt
and
From (5), (7) and (9), after simple algebra we obtain
b= 2-P(a2 - I)&-2(6’&,,),.
(8)
110
L. Gajek
et al. I Statistics
& Probability
Letters
28 (1996)
107~110
On the other hand, the definition of b combined with (5), (7) and (8) yields l/2
b=2
I0
xP(t)dt
= 2
(11) Eliminating the constant a from (10) and (1 1), we find that b = (2e(~))-~(S(p,lQ(~))(l where S(P) =
I.I tp o m
Making the substitution S(P) = (llP)K(l where B(s,z)
- S(P>/Q(P))-~:‘~>
dt,
and
Q(P) =
Is
1 ___ 0 Jr-rp
(12)
dt.
u = 1 - tJ’, we get + PYPY ;h
Q(P) = (llP)RlPT
;I>
is the beta function. Finally, from the last two formulas and (12), we obtain that
which completes the proof.
0
References Chung, K.L. (1949), An estimate concerning the Kolmogorov limit distribution, Trans. Amer. Math. Sot. 67, 36-50. C&go, M. and Rtvdsz, P. (1981), Strong Approximations in Probability and Statistics (Academic Press, New York). Csorgii, M. and L. Horvath (1993) Weighted Approximations in Probability and Statistics (Wiley, Chichester). Finkelstein, H. (1971), The law of the iterated logarithm for empirical distributions, Ann. Math. Statist. 42, 607-615. Gelfand, I.M. and S.V. Fomin (1963), Calculus of’ Variations, Revised English ed. (Prentice-Hall, Englewood Cliffs, NJ). Lenic, A. (1994), Projection methods for nonparametric density estimation, Ph.D. Thesis (in Polish), Polish Academy of Sciences. Smimov, N.V. (1944), An approximation to the distribution laws of random quantities determined by empirical data, Uspehi Mat. Nauk. 10. 179-206.