UHlVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill,
N~
C.
ON TEE PROBABILITY OF LARGE DEVIATIONS
OF FAMILIES OF SAMPIE MEANS
by
J. Sethuraman
June 1963
Grant No. AF - AFOSR - 62 - 169
Results concerning the exponential convergence to
zero of the probability of large deviations of the
sample mean have already been obtained in the
literature, for instanc~in Cramer ~5_7, Chernoff
£4_7, Bahadur and Rao L 1_7, etc. We establish
similar results about the exponential convergence
to zero of the probability of large deviations of
the sample distribution function, and more generally, of families of sample means.
This research was supported by the Mathematics Division of the
Air Force Office of Scientific Research.
Institute of ~istics
Mimeo. Se~~:N6. 368
ON THE PROBABILITY OF LARGE DEVIATIONS OF
FAMILIES OF SilMPLE MEANS
I
by
J. Sethuraman
University of North Carolina
= = = = = = = = = = = = = = = = = = = = "= = = = = = = = == = = = = = = = = = = = =
1.
..
General
.,.. .... Summary
.. ..
~.....
~
Results concerning the exponential convergence to zero of the probability of
large deviations of the sample mean have already been obtained in the literature
for instance" in Cramer
£2.7"
Chernoff
£}j;,7"
Bahadur and Rao
£1:7"
etc.
We esta-
blish similar results about the exponential convergence to zero of the probability
of large deviations of the sample distribution function" and more generally" of
families of sample means.
2.
Introduction and Summary:
Let
(-1-2,
S " p) be a probability measure space.
Xl(W)" X2 (w)" •••
Let
be a sequence of' ~measuJ:'able random variables with values in
pendently and identically distributed with camnon distribution
X
which are inde-
IJ.(.).
Here ~ is
a separable ~omplete metric space and ~~iS the class of all Borel subsets"
J.I.(n,~,.) is the sample distribution function of
probability measure that assigns masses
Let
f(x)
% at
X ('il1)" ••• X ("')1 namely, the
l
n
Xl(i~)"".Xn('~).
each of the points
be any real valued measurable function on "{ with
(
.J f(x)
IJ.( dx) <
co.
Then the strong law of large numbers due to Kolmogorov
states that
••• + f(Xn (to»
n
-
f(x) lJ.(dx)
.->
0
I.
i
=
1
•
~his research was supported by the Mathematics Division of the Air Force Office
of Scientific Research.
j
Again, >OIlen
2
~(ax) < .. , for all
etf(x)
t,
tha abova result has been improved
upon as follows" by Cramer (1938)" Chernoff (1952)" Bahadur and Rao (1960)" etc~
1:n
f(Xl(W»
log P
f(x)
n
---->
where
-
I
+ .' •• + f(Xn(w»
log
~(ax)
p(f,!)
0 < p( f '" ~... ) < 1 and will be defined in (14). We interpret this result
'
-
in words by saying that the probability
o~
large deviations of the sample mean
tends to zero exponentially.
1t,
Let
F(n1-"'-'x)
k
= R '" be the Euclidean space of
k
dimensions.
represent the distribution functions corresponding to
""
(
Let F~)
and
J.1(. ) and
Il(ru"u)" .) respectively. The Glivenko-Cantelli theorem states that
(3)
P {"':
s.;: I
F(n,w,o.:)
-
F(;;)
I ---> 0 1
=
1
We now ask the question whether the probability of large deviations of the sample
distribution function 'Wi 11 tend to zero exponentially.
We answer this question
by the following theorem.
The~rem
1.
For e
>
0 •
)
\
F(~) I ~ e f -> log p*(f,e)
log P ( W: sup \ F(n"u>,,~) nIx,
1:
(4)
.-
where
0 ~ p* (F"e) < 1 and - p*(F"e)
Now we can interpret F(n"w"x)
...
will be defined in
(26).
as the sample mean of a certain real valued
.~."
function or as the sample probability measure of a certain subset of )( • Corresponding to each of these interpretations we can generalise the above theorem as
The proofs of all these theorems are given in section 4.
follows-.
Theorem 2.
Let
3- be an equicontinuous l
real line.
Let
g(x)
class of continuous functions from
be a continuous function such that
~or a definition see section 3.
I f(x) I
*
~ g(x)
to the
for
•
each f
in
'J
%log P
I
and such that
etg(x)
~(/!x) < ~
fIll'f;1' \l~(x) ~(n,<JJ,J x) - l
f(,,)
----> p( J
where
p(
J
,e:) will be defined in (15)
3
for all
~(ax)!
I
t.
Then for s
> 0,
> s
e:)
4
Theorem 3.
-
Let :;.. be a class of continuous :f'unctions from:x.
the uniform convergence on compacta l (u.c.c) topology.
for each each f
in
3 .
.
For each a
--
the set
Then for
e: > 0
k
in Rand
that is compact under
~
Let
f
in
~
f
-1
f' be a non-atomic
let
A(f , !:) be
I
f
where
1 log P w: sup
n
\ 1r,J0 < p* (e:) < 1 and
p*(e:)
will be defined in
(26).
These results are allied to those of Sanov £lJ::7 and could be possibly deduced from them..
However elementary and straightforward proofs are presented
here.
Section 3 deals with the necessary preliminaries and some well known lemmas.
Section 4 gives the proofs of theorems 1,2 and 3 and section 5 contains some
of their applications.
A result related to theorem 1, giving some upper bounds to the probability
on the left hand side of (4)
may be found in Kiefer and vlolfowitz
~or a definition see section 3.
L27.
Weaker
forms of theorems 2,;,4 and 5 have been obtained by Rao ["12,7.
A result ex-
tending theorem 4 but weaker in other respects may be found in WolfoWitz L:l~7 •
Theorem 6 extends the results of Blum
L:i7.
Preliminaries~
;.
A sequence of measures
\I,
\In:=.:t \I
t \In J on
(1:,'D)
is said to converge weakly to
in symbols, if for every bounded continuous function
h
on~· ,
!
) hex)
In such a case, for any
\Ie ax)
•
6 > 0, there is a compact set K int.. such that
\I n (K)
> 1 - 5 for n = 1,2, •••
\I(K)
> 1 - 6. This result may be found in Billingsley
and
•
.r~7.
~.-!
A family of functions
there is an
r
j-
such that
If(x) - f{x') \
where
d
is the metric on
< 6 for all f
l .
in
J- whenever
i
A sequence of functions
f
to converge uniformly on compacta (u.c.c) to a functions
to
f{x)
J- bounded by a
compact under the u.c.c. topology.
In this case for any
f
in
5
.
<r
~ on ~. is said
fn{X)
6 •
g
converges
l .
is conditionally
> 0 and compact set
{fl , •••
f , there is an index~'such that
/1
K'
)
7, states that an
function
*'" , we can find a finite collection of functions
such that for any function
.1. (.\.
d{x,x t
for each x and the convergence is uniform on each compact s"t.!n
equicontinuous family of functions
sup
n
f, if
A theorem due to Ascoli, for instance ~ee Kelley L:~7, Chap.
K in
6 > 0,
is said to be equicontinuous, if for every
f m } in
J-
5
Let
110
(5)
= { OJ : jj(n" OJ" .)
1·
--9 ~(.)
An interesting result due to Varadarajan flg7 which is a generalization of the
Glivenko-Cantelli theorem states that
f6(r»
~ -0
(6)
=
1.
5> 0
A conclusion that we draw from this is that for each
there is a cOIIlJ?act set KOJ
jj(n"
(7) (
Let
g(x)
be
-t
-
1 - (,
w)
->
1 - 5 •
aDy
in
i-2
'
0
such that
w, Kw) >
jj (K
and
in
and OJ
.n
= 1,2" •••
measurable function with
I
g(x)
~(.tx)
< ...
Let
(8)
•
(1)
The relation
can now be rewritten as
Another conclusion that we can draw from this is the following.
and OJ e
1-11
' there~tsa compact set K
w such that
) g(x) jj(n" w, dx)
(10)
~. -
and
I(
'AJ
For any function
(11)
E(f)
< &, n
J g(x) jj( dx)
~-K
<
f ~e define E( f)
f.
= } (x)
jj( dx) •
&
= 1,
•••
•
by the relation
For each
0
>
0
e
6
Let
tf
be a function for which E(e ) <
f(x)
CD
Hl(f,t,e)
=
E(exp (tf - tE(f) - te)
H (f,t,e)
2
=
E(exp (tf - tE(f) + te)
(12)\and
for all
t.
Define
for
e
> 0,
•
Define
Pl(f,e)
=
P2 (f,e)
=
(13) \ and
inf Hl(f"v,e)
t > 0
inf
t<O
H (f,t,e)
2
Define
e
For any class of functions
is in
J '
'j- such that
tf
E(e )
<
co
for all
t
whenever
define
p(-l ,e)
J
=
sup p(f,e)
fej-
•
The following are well known and are given here only for completeness.
(16)
min
O<t<T
- -
where
Xl'.
T
is finite
(ii).
P1. f
- E(f)
5
e} =
1
and
f
=
Pl(f,t,e)
and given any
Q
then given any
(18)
P
t
f - E(f)
=
e
1
7
> 0, there is a T such that
Q
> 0, there is a T such that
min H (t,t,e)
O<t<T 1
< Pl(f,t,e) + ~
and
This follows from lemma 1 and a similar result for H2(f,t,e).
Lemma 3. If
is a class of continuous functions that is conditionally under
Proof:
3*
the u.c.c.
topology then the function
p(
3-,
e)
at each e > 0 •
Proof:
Let
and
5 > O. We note that
Hl(t, t, e-5) > Hl(f,t,e)
for t> 0
H2(f, t, e-5) > H2(f,t,e)
for t< 0
-
-
Hence
p(f, e - 5)
Thus
->
p(f,e)
for each f.
p(.j- , e-5) > p(-.=.! ,e).
-
Hence
(19)
lim '"off
5->
0+
p(
j ,
e - 5) ~ p(:f, e) •
is continuous from the left
8
Now, let
e
n
be a sequence of numbers tending to
f f n J of functions
find a sequence
Since
J
is compact, there is a
city, that converges to
f
in
J-
e.
For any Q > 0, we can
such that
subsequen~e
of·f fJ, say {fn ~ itself, for simpli-
uniformly on compact sets.
uniformly Qn bounded intervals to t.
Hence
Further
and
<
for each finite
T.
Again given Q
of lemma 2.
We thus obtain
(21)
lim··)u[.
r
for arbitra:ry
Q.
min
O<t<T
<
p(f ,e)
n
n
Combining
H (r , t, e )
2 n
n
> 0, we can find a T to satisfy relation (18)
p{f,e) + Q ,
(21) and (20) with the fact that p(r,e) < p( f,e)
we obtain
(22)
11m41Jt.,
p(]-, e)
n < p(
J-,
e) •
(19) and (22) establish lemma 3.
We now require some 1'roperties of
tain classes or functions.
e
p{ f
= l}=
1"
Then
p(f,e)
Let
f
p(]-,
e) where
>,f
is restricted to cer-
be the indicator function of a set and let
depends only on l' and is written as p*" (1',e).
We can now reqrite result (2) as follows.
If Xl(f,:J), X (w),...
2
is a sequence
9
of independent and identical binomial random variables with mean P, then
- p
We can find an explicit expression for
I>-
e '. ---> log P*(Pl e ).
J
P*(Ple) from (14) as follows.
where
(l-P
)
1- P - e
l-p-e
.
••• 0 ~ P ~ 1 - e
l-e-p~ 1
and
t1(Pl e )
Let
k
F(x)
be any distribution function in R • We define
.~'.'
(26)
*
P (F , e)
f
=
sup
.....x
p*(F(x)
,... 1 6)
land
We now state a few results concerning the function
Lemma 4. (Hoeffding)
If
e > 0
and 0
< P < 1/2, then
P*(PI S) _< exp
2
_r _ l-2p
2 6
and hence,
(28)
P
~im
0
->
P*(PIS)
= o.
log
l-p
P -
7
P*(Ple).
10
Proof:
Lemma.
This result "!DB.y be found in Hoeffding
(1963, relation (2.4)J
5.
The function
Proof:
p*(F,e)
is left continuous at each e > 0 •
The proof is elementary and proceeds through steps analogous to lemmas
1, 2 and 3 and 'Will therefore be omitted.
We now proceed to a leW- WUlWimOw. lemmas which are connected 'With uniform
convergence of distributions.
Lemma 6.
J
If I) f n is a sequence of continuous functions converging to a function f
uniformly on compact subsets then
Il
f-l....),.
II
n -/ ,..
f- l
and , moreover, if,.. f- l is non-atomic, then
II
sup
a.
where A(f,a)
Proof:
I ~(A(fn,a»
- Il(A(f,a»
\
~
0
is the set
The first part of the lemma is well known.
The second part is eq-qally
well known and is usually referred to as Polya' s theorem.
Lemma 7.
~-7
Let .j- be a class of continuous functions that is compact under the
u.c.c. topology and
(31)
f
p f -1 is non-atomic for each
I'
in .J-:J.. • Then as
5 -> 0
sup
e J
where f + 5 is the function f(x) + 5 and A(f,a)
is as defined in (30).
11
e
The lemma. is easily proved by contradiction using lemma 6.
k
Lemma 8. Let
= R and
be the class of linear functions from
Proof:
t
'l'
of norm unity.
Then as
sup
~(A(L
sup
a
L€"l
->
8
into R'
0
+ 5, a»
-
~(A(L,a»1 ---> 0 •
This is immediate from lemma 7 since
Proof:
' *:.
~
is compact under the u.c.c.
topology.
Whenever
,'fk
space.
1 = nk
and
L in
cL
we will sometimes call A(L"a)
will denote the class of all sets that are formed by the intersection
\,~ \
of m half-spaces.
represents the norm of .~ in the Euclidean space.
s(~ , r) is the closed sphere 1~: I~-! I :s r J'.
Set), r) = f~:! € D,H~-!\:S rI·
For any subset D in
k
e
a half
We will also need some properties of convex sets in R.
t ,
For the general
properties of convex sets" the treatise of Eggleston (1958) is an excellent
reference, but Section 4.1 of a recent paper by Rao ["12,7 covers the material adequare for our purposes and we shall reproduce that section with slight alterelltions.
Let
,
C be a convex set with non-empty interior.
Let a
be an inner point
(.~';. f:) "
'!'"
Then the guage function't:lH(~), oot C with respect to .~
of C.
is defined as
follows:
() =
H,~
)
x-a
inf l X: a + - -;:
It is clear that if H(~)
C=
i ~ : H(!)
of C = [.::
<
H(~)
I} and
= l}.
€
",I
G( •
is the gauge function of C, then CO = interior of
C = closure of C =
f ~ : H(~) ~l} and the
boundary
The following properties of a gauge function are well
known and can be immediately deduced from its definition
(i)
If H(x)
.....
is the
(a) H(x) > 0, and H(x)
rv
c
-
""'"
> 0, and (c) H(~~!.)
=0
g.-". of C 'With respect to the origin then
if and only if x
:s H(~) + H(l)'
=0
, (b) H(cx)
~,..
= cH(x),
........
for all
for each pair ~'.!' Conversely any function
H(::> with the above properties is the gauge function of the convex
setf~:H~) :s
l}'
12
(ii)
H(~
Suppose that
origin and that
C:
S(O"r)
""
For any convex set
is the gauge function of C with respect to the
Then for each x,
C.
H(x)
<
•
'"'-
"...
C, the inradius of C, denoted by
r(C), is defined to
be
r(C)
= suptr:
S(.=::, r) C" C, for some!:, in C}
Then it follows from definition that
<
interior and thatr(C)
,
H(~)
if C is bounded.
(iv)
associated with C for which
in Rk •
for all x, y
We shall call this the primary gauge function of
If C is a convex set with inradius
function
if and only if C has empty
If C is a convex set with non~empty interior then there is a gauge
(iii)
function
ex>
r(C) = 0
•
e.
< 0: then there is a linear
L such that
ec
r-
~ x: . a
L
5
L(x)
< a +
k
denotes the class of all closed
Let K be any bounded set in R •
> 0, ( (K, 1,
0:)
closed convex sets contained in K and of inradius
>
convex sets contained in
-( (K) in section
K.
for
0:
denotes the class of all
0:.
c: (K, 2,
0:).
=
l
(K, 1" ex). The final results about convex sets that we shall use
4
can now be stated as two lemmas.
Lemma 9.
The class
j.;/
.?
of primary gauge functions" H, of the elements in
is compact under the u.c.c. topology.
set has
IJ.-measure zero then
IJ. H- l
[ (K, 1,
Further if the boundary of every convex
is non-atomic for each H in
Ji .
0:)
1,
e
Proof:
The proof is immediate fram property (iii)
Lemma 10.
Proof:
As
a ---> 0 ,
We note from (iv) that
s(c
,a) c:
{ x : a ..
a<
L(x)
~
a +
,a.J .
The lezmna follows·fram lemma 8 •
Again let
i = If·.
C~ 1
Let
be the
the
k
Yi
< xi,i = 1, ••• k, then
.! E:
A.
(~j' j
Let
!
= (y l~ ... Yk)
= 2, 3, ... ,
~ ,
2,k"'l classes of sets which can be obtained by reversing, one at a time,
inequalities occuring in the definition of
For two classes of sets
D {\
A each of which
If ;S = (Xl' ..... , ~) E: A and
possesses the following propeX:-y •
is such that
be the class of sets
J)
and
t
E Where D is in!J and E
let
in
a
=
the class
013
1.
jJ@f! denote
(l
..... .
ffl
Let lX
2k
Y
-"
(J'j.
~~~s
Let
•••
the
I
product I containing a finite number, say m, terms.
Lemma 11.
(Blum)
Given any
5
in
a
Let
IJ.
be absolutely continuous w.r.t. the Lebesgue measure.
> 0, there is a finite class of sets (I.) (&) auah that for each A
there exist two elements
B
l
and B2,
in
03 (5)
such that
•
Proof:
The proof of this lemma follows fram elementary considerations of the
properties of the sets in
a
and may be found in Blum 1'2.7
•
14
Lemma 12.
Let
for each
be absolutely continuous with respect to the Lebesgue measure.
Jl
8 > 0, there is a class of sets ~*(5) such that for each A in
there exist sets
Proof:
and B
2
B
l
in
(& *(5)
Then
(t*
such that
This follows immediately from lemma. 11 and the definition of
.- * .
ct.
4. Proofs of TheoreIIlS 1, 2 and 3 ~
Proof of theorem 1.
Let
G (x?, •••
l
tiosn of F(x).
' .....
G (x)
be the right...contintious marginal distribution tunc-
k
For any
8
(~l) points
> 0, we can choose k sets of
(~o' ~l' •••• ~)
corresponding to the k marginals so that
(
(34)
i
_
0:>
,)
l
=1, •••
Any point
=
x
io
G(xis - 0)
k
and
.:5, =
s
< xil <
-
G(x _ )
is l
= 1, •••
(xl' •••
< x.
•••
~m
<
+00
5
m.
~) in Rk will satisfy the relations
-< xi s i
for some integers
=
sl'''.' sk'
i
1 ~ si ~
= 1, •••
m, i
k
= 1, ...
analysis then shows that
sup
x
I
F(n,L.; ',,3)
-
F(~) I
,..-...,
~ sup IF(n,!.,;"
:te £*
J.S*) -
F(x*)
I
+
0
k.
A straight-forward
15
where
=)x*:
]L*
l~.
:A
with 0 -< i s<m
x*
--
= (xli
..
1
s = 1 , .•• k •
1
' •••
J·
Also from (23) we have
~
(57)
log P
t"" \
F(n, '"
->
log
~}
- F(:t}
I~
0 -
81
p*(F(X*), E - 6) < log p*(F, E - 5) •
-
""
(36) and (37) yield
(38)
Since
~
lim sup
5
I
lOgP(" : s? F(n,
lll,
~}
-
F(!E}
I ~ o} c:
log f1*(f, 0-8}.
> 0 is arbitrary and p*(F,E) is left continuous from lemma 5 we have
-
Also
p, \ ,,: ~up
\ F(n,
....
for any x
.-
k
in R •
(40) lim:1n:f
i
w, ,,} - F(:,:} I ~ 01 ~ p {w: IF(n, '" ~} - F(~} I ~ 0j
Thus usJilng {2}~ and noting that
log P{" : s;( F(n, w, J!:} - F(:::)
""
(39) and (40) prove the theorem.
Proof of theorem 2.
,
x
is arbitrary we obtain
-.;
I ~ 0] ~ log f1*(F,oj.
16
It is enough to show that
.1
log P
n
1fe~. J1. ,-)( f,
.l -
Choose and fix a
0
satisfying relation
(10)
e)
>
1
- ..->
~
For any
O.
tn
Ul
log
.-.,
j-,e).
p(
1-1 ,
1
there is a compact set K(l)
0·
:;f-
(10), namely,
\ f(x) Il(n,o,dx)
t -K(l)
and
~
<
f(x) Il(dx)
0, for all
f
in
-K
'l
ill
:t:
Further since }- is conditionally compact we have a finite collection {fl , ••• f }
m
of functions in
J
such tlnt if
f
is in ]
there is an index i
~~p \ f(x) .. fi(X) \ < 0 •
·x c 1<\.))
Relations (42) and (10) yeild the inequality
(42)
sup
f e
3:
I)
(f(X) Il(n, w, dx)
~
Jf(X)
..
sup
l(fi(X) Il(n,
l<i<m
Il( dx)
w, dx) -
)fi(X) Il(dx)
Thus
(44)
U ..rl(f,e) c:
1t J
m
U
i-1
i:;l
(fi ,
e-
6 5) •
Relation (2) can be rev~itten as
-'.
ln
log p('-) (f,e»
.l -
--> log p(f,e) •
I
I
+
60
such that
17
From (44) and (45)
(46)
lim sup
vIe have
1:
log
n
Pi) U ,[1
(f"e:)
.
'jl.
~l<i<m
max log
fe:j-
-
-
< log
Since
5
>
0
is arbitrary" and
p(]-,
e:)
p(fi , e: - 6 0)
pC
J-,
e: - 6 5)
is left continuoaa
•
from lemma:; we
have
(47)
11m sup
Again,
~
log
p) L~ -Cl(f,e:) }
LJ..j- ..\.'-)(f"e:)
..,
(-)(1',6)
~-' - 11m inf
1:
n
log P}l
•
J
l) r) (1',
fe: :;..\. -
The theorem now follows from
J , e:)
for each l' in-.:J • Using (45), we have
1'6
(48)
< log p{
tfe: J~
e:)}
> log p(
-
'3_ ,e)
"J
•
(47) and (48).
Proof of theorem :;.
Define
o >
O.
'{-lo as in (5). From (6) we have
11(;l
Then for each w in
PJC..rl o )
= 1.
Choose and fix a
there is a compact set K in"* satisfying
w
reaation (7)" namely"
5
(7).}
!-L(n, w, Kw)
land
> 1 - 0
!-L(Kw )
Let A(f,;:)
r
•
.
=.1. x:
flex) ~ a l , .. • fk(X)
~ ~
2_ .
J'.
under the u.c.c. topology, there is a finite collection
members of
j- such
sup
xe:K
that for each l' in
w
1 f{X)
-
f i (x)
J- there
I<
5
•
j
Since
[1'1'
is an index
is compact
m } of
such that
... f
i
18
Using this fact and relation (7) we have
in
for each ().~
(-)
-
•
-0
Now choose and fix any
sup
where
sup
a
.....
.3
f£
Q
> O. Using lemma 7 we choose 0 so that
~»
} J.L(n,\r), A(f,
. . J.L(A(f..~» \
'
<
sup ;max
l~
iSn
i~j~
sup \ \.l(n,{.o), A(f +
i
a
¢j
0,
¢j
0,
~»
(\J
... Jl(A(f +
i
};»
I+40+ 2
¢1 = 1 and ¢2 = -1.
If
'Y
is any function from ~'into
f
0f'\ k
with non-atomic induced distribu-
tion, we have from theorem 1 that,
1
(50)
n
where
log
p* (e)
is as defined by (26) •
.
Since
(51)
Q •
f
lim sup
i
+ ¢jO
l
n
has non-atomic induced distribution we have
log P L-.\.): sup
t
f£
sup
a
"'"
< log p* (e - 4 0 - 2 Q) •
19
Since
is left continuous by lema. 5 we can replace the right hand side
p*(e}
of (51) by log
p*(e}.
Now
sup
fe .3-
>
for each l'
in
sup \
a
j- .
>
e
~(n,,;.'J 1 A(f,~»
..
~(A(f,,~»
\
This together with relation (50) gives
1Qg
p*(e)
•
The theorem follows from (51) and (52) •
5. Some Applications of the Results of section 4:
We now state and prove three further theorems which are applications of the
~.
(53)
where
.
log
f( m
pl
(;.J:
sup \
AE~m
~(n"lu"
A) .. IJ(A) \
~
e} .... ~ log p*(e)'.
.
is the collection of all sets formed by the intersection of m half...
spaces.
Proof:
L
=
This theorem is immediate from Theorem 3 and the observation that
all linear functions of norm 1
is compact under the u.c.c. topology.
20
Theorem
5.
Let (
e
,-
C c:
set
be the class of all closed convex sets.
, where 'bd C ::: boundary of C.
~(bd C)
=0
for each
Then
•
~
(54)
Let
log P{lti.:
sup
ccf....
l~(n,aj,c)-~(c)l >
el->log
p*(e)
•
Proof:
Choose and fix a
I)
>0.
We can find a bounded closed convex set K
l
in
such that
We divide
Linto three classes". for some
ex, wi,th
1 >
ex > 0, which we will choose
later.
)'""
\,... ~_ =
where
and
Ki
class of all convex sets
is the complement of
n Ki
C with C !
not empty
,
Kl •
Let K = S(K , 1). Then trivially
l
Let
C"c.:
e
(K , 2)
l
ex) • Then ex, with
1
so that
~(C)
Thus
<
I)
•
> ex > 0, can
be chosen from 1ennna 10
21
NovT S(C ,ex)
member of
is convex, has inradius
(; (K, 1, ex).
> ex and is contained in K and so is a
Thus
Combining (56) and (58) we have
C c:
Let
t'. * •
--
Then
+ fl(nll,;.l , Ki)- fl(Ki> + 8
Here
C
n
K
l
is convex and is in
e (K).
•
Thus combining (59) and (60)
we have
~
sup
\ fl(nl ..4J , C) - fl(C) , + ifl(n,wl Ki>
CCl(K,l,ex)
Consider
Ii
u.c.c. tOJ.'ology from lemma 9.
fl(Ki> , + {) •
C(K,l,ex). This
Each C C-l (K,l,ex) is
the class of primary gauge functions of elements of
class ie cOIn];lact under the
-
r:,:,
the set
,
H(~) 5
1
22
1
H(~)
for some
in
j-{
and from the conditions of
the theoremS H has non-atomic induced distribution.
(61!)
lim sup
~ 10e pi""
Thus from theorem 3 we have
I ~(n,IiJ,C) - ~(c) \
sup .
C c (K,l,e:)
r
1
~ ~
< log
p*(~)
•
Again, from relation (23) we have
(63)
;
log p-{ (J):
~(n,lA:l.,
\
~(Ki> t
Ki> -
~
From (61), (62) and (63), for each fixed
(64)
lim sup
~
log
pt~UJ:
sup
Cc
t:
log
Q,
>
Q}
p* (~(Ki)'
with
I~(n,~, C) -
From lemma 4 we know that for each Q
for sufficiently small
>
£ -
0
Q) •
>
Q
~(C) I ~
0, p*(p,Q)
->
>
0,
we
have
£ j"'
° as
p
->
0.
Hence
0
p*( £) is left continuous from lemma 5 we can replace the right hand
.
side of (64) by log p*(€). The corresponding inequality for the lim inf is
Again since
trivial.
Hence the theorem is prcved.
Theorem 6.
Let
Cl._*
IJ.(.) be absolutely continuous with respect to the Lebesgue measure.
be as defined in (33).
Then
Let
'"t..):
e
Proof~
The proof of this theorem follows immediately by the application of lennna.
12.
6. . Acknowledgements.
The author wishes to thank Professor C. R. Rao of the Indian Statistical
Institute for suggesting the problem and for the thoughtful conversations, and
to Professor W. Hoeffding for several helpful guidances and remarks.
7.
References.
I)}
Bahalihl~, R. R. and Rao, R 4I Ranga
(1960)41 On deviations of the sample mean,
Ann. Math. Stat., 31, 1015-1027.
~~7
Billingsley, P. (1956).
The invariance principle for dependent random
variables, Trans. Math. Soc.,
~~7
Blum, J. R. (1955).
~,
250-268.
On the convergence of empirical distribution functione,
Ann. Math. Stat., 26, 527-529.
~!±7
Chernoff, H. (1952).
A measure of asymptotic efficiency for tests of a
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...
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/
)
Cramer, H. ( 1938.
Sur un nouveau theoremeiimite de la theorie des proba\
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/
i
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~§l
Eggleston" H. G. (1958).
Convexity, Cambridge tracts in mathematics and
mathematical physics, No. 47.
.L17
Hgeffding, W. (1963).
Cambridge
~§.7 Kelley, J. L.
127
Press •
Probability inequalities for sums of bounded random
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.
Univ~
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.2.§, 13-30.
(1955). General Topology, Van Nostrand, New York.
Kiefer, J. and Wolfowitz, J. (1958).
On the deviations of the empiric
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§I, 173-186.
£J.g,7
Rao, R. Ranga (1962).
Relations between '\-Teak and uniform conver8~fice of
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£1~7
Sanov, I •. N. (1957).
42
<.mt),
On the probability of large deviations, Mat. Sbornik,
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£~7 Varadarajan, V. S. V. (1958).
On the convergence of probability distribu-
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£1'2.7 Wald, A. (1947). Sequential Analysis, John vTiley Press.
£1!i7 WolfoWitz, J. (1960). Convergence of the empi:t'ic distribution function on
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