INVARIANCE THEOREMS FOR CENTERED
SEQUENCES NORMED BY SUMS OF SQUARES
by
A.G. Mucci
Department of Biostatistics
University of North Carolina, Chapel Hill
Institute of Statistics Mimeo Series No. 1150
December 1977
INVARIANCE THEOREMS FOR CENTERED
SEQUENCES NORMED BY SUMS OF SQUARES
by
A.G. Mucci
Department of Biostatistics
University of North Carolina, Chapel Hill
ABSTRACT
t
n
Let
L -+
n
00
Sn
a.e.
=
f k , Ln
Then
where
=
S
n
Ii;
Ifn I ~ 1, E{fn Ifl,fn- I}
=0
and
d
-+
n(O,l)
and
lim
Sn
n 12L log log L
n
n
=1
a.e.
The methods leading to these results essentially involve strong inequalities of independent statistical interest.
These inequalities are modi-
fications of results due to Freedman.
AMS 1970 Subject Classifications:
Key Words
&Phrases:
Primary 60F05, 60F15; Secondary 60G45
Martingales; Invariance Theorems; Iterated Logarithms;
First passage.
-2-
Introduction
Let
(Q,B,P)
of sigma-fields
is
be a probability space supporting an increasing sequence
{B}
n
B -measurable,
n
E{fn 18n- I} = O.
and a sequence of random variables
i.e., where
is adapted.
{f ,B }
n n
{f}
n
Assume
where
Ifn I
:$
f
n
1,
Let
n
S
n
n
=!1
L
Then, under the assumption that
n
Sn d
(A)
-+
Ii:'"
n
= l1
L
n
fk
+
f
2
k
a.e.,
00
we have
nCO 1)
'
and
S
lim
n
n 12 L log log L
n
n
(B)
a.e.
=1
Result (A) is a special case of the following:
a > 0
X
a
be the process whose paths are the linear interpolations of the points
2
(a L ,as).
n
(C)
where
n
and let
Then
X
a
W is Brownian Motion.
d
+
W as
a
+
0
I obtained these limit theorems by combin-
ing results from Freedman [,] and Freedman [].
(D)
Let
y
d
a
+
W as
a
+
In Freedman [ ] one finds
0
and
(E)
lim
n 12 T
n
Sn
log log T = 1 a.e,
n
e"
-3~
e
t
2
n
T = E{fkIB _ } and Tn + 00 a~e"
and where Y is the process
k l
n
a
2
whose paths are linear interpolations of (a T ,as ), In Freedman [ ] one
n
n
finds
where
C
-E..
(F)
M
n
n
where
n
= L gk'
C
n
1
+ 1
Mn
= L E{gkIBk_l}' o ~ gn
approach in deriving (B) and (C) from
for
~
1 and
M
n
1
+
00
a •e .
My
(D), (E), (F) consists in using
The body of the present paper is a combination and rearrange-
ment of Freedman's papers, with statements of theorems and methods of proof
modified either mildly or severely. My principal concern has not been the
limit theorems but rather the inequalities which lead to the limit theorems,
for these inequalities given considerable information concerning the distribut ion of partial sums in terms of the partial sums of squares and may therefor prove useful in statistical contexts where independence and knowledge of
conditionals is not assumed.
e
~
1/3
For example, in Section (lV)1 prove that for
and reasonably large
p{Sn
~(l+e)r'2
N:
L > N} ~ ~-r=_3
__
7 2
some
Ln log log Ln
e/
n -
(log N) a
Such inequalities are examples of the general form
p{s
n
where
~
<P,<P
~
lj>(L ) ,
n
some
are specified functions.
L
n
~
N}
~
~
<P(N)
The latter type of inequality is the
modification called for, when independence and knowledge of conditionals
is not assumed, of the inequality of type
"-
-4-
p{S
where
S
n
~<t>(n)
n
,
some
n ~ N}
is the sum of centered i.i.d, 's with variance
Inequality
(*)
1.
occurs in my proof that
lim
nl2
Sn
s 1
L log log L
n
n
This proof differs considerably from Freedman's proof that
lim
n 12 T
n
Sn
-::-----,=__ s 1
log log T
n
and is an instance of a proof motivated by possible statistical applications, namely such as are provided in Darling, Robbins [
].
Although
statistical applications are not directly considered in the present paper,
a path toward such applications
j
s indicated by the inequalities and
will be developed in a subsequent article.
The result (C) is not new - it appears in McLeish [ ], Rootzen [
and Drogin
~;],
]
although these sources were unknown to me prior to com-
pletion of the draft of my paper. In fact, all these sources present a somewhat more general result.
The methods used in these papers, however, are
not appropriate for the determination of strong inequalities, and it is
strong inequalities which form the object of the present paper.
Result
(8) appears to be new.
Finally, a first passage time is considered in the remarks at the close
of Section (IV).
The methods of this paper are shown, in this particular
instance, to be useful for finding upper bounds on expectations of passage
times.
These methods will be extended in a later paper.
-5-
(I)
Results for Positive Variables
Let
{g
n
variables on
for all
n.
B , n ~l}
1
n
(Q,B,P) •
be an adapted sequence of non-negative random
Por convenience, set
B = {¢,Q}.
O
Set
n
C
n
=
t
1
gk
n
M
n
=
t
1
E{gk IB k_1}
Set
F (A)
= 1-e -A
G(A)
= e -1 ,
A
A
~
0 •
We define
~ =
e
=
e
R
n
Proof.
AC -G(A)M
n
n
{Freedman []}
Theorem (1.1).
{~,Bn}
F(A)M -AC
n
n
{R , B } are supermartinga1es.
n n
and
We show that for
0
~
g
~
1
and
A c B,
we have
and
Since, for
0_
x
~
0;
1-e
-AX
x
decreases in
x
Assume
g ::; 1
n
-6-
while
e AX "" 1
increases in
X
x ,
we have
e->.g ~ 1 ""F(>.)g
e
and
~
Ag
1 + G(>') g •
Take conditionals on both sides and use the inequalities
-X
1-x:~e
1 +x
~
00
Corollary (1.1).
X
Q.E.D.
00
Let
Then,
Moo =
Proof.
e
a.e.
00
<~>
C00 =
00
e
a.e.
By optional stopping, for each proper stopping time
F(A)M -AG
1
~
1
~
J
T
c
•
T:
T
and
)'G).-G().)M
Suppose
M00 =
00
a.e.
T
T
Define
T =
Note that
J
e
is proper and
first
n
as
M
n
a ....
~
T ....
00
1 ~
J eF(A)a-Ab
a
00.
Fix
b
>
O.
From the first
inequality above we have
{CT~b}
so that
e"
-7-
and letting
C00
=
00
a.e.
a
~
p{C 00 <; b} ::; 0 ~
we have
The converse is
Corollary (1,2) ,
Let
(a)
Coo
p{:n
n
(b)
_
00
p{:n
n
(c)
p{:n
n
(d)
p{:n
n
~I
=M =
+e
some
~1- 8
some
C
n
Cn
~ t~)
~
e
8
- -
~ t}
~
some
M
~
~1_e ~
some
M
n
S
T
p{C
(f)
T
~
T
M
T
~
e
2
8
2
~
a}
(l+E)a}
S
~
e
2
2(I+E) a
-E
2
e2(1+E) a
Let
p = first
o
T
"-
n:3 Cn
= first n:3 Cn
=
first
{
o
n€
:? t
~
0~
t
Proof.
(a)
~
2
and for all proper stopping times
(l+E)a, M
~ a,
t
2
8
-t
e
~
t}
n
2
8
-t
~ t} e
~ 1+8
p {C
Q.E.D.
2
8
- -3 t
-E
(e)
consequently
in the same manner.
Then for all fixed
a.e.
00
00
a>O, E>O
Further, for all
~stablished
b~
all finite
s > t
s
[p,o]
if no such
with
n
~ ~
occurs
(1+8)C
n
T:
8 ~~:
-8-
Then
1
~
e
I
FC~)M't"'~C't
(F (A) (1 +e) "'A) C't
~
e
I
{'t<n}
{'t<n}
provided rO.)
~
~ e(F(~)(l+e)",~)t
~
_1_
l+e·
Under such circumstances
The right side minimizes for
~
= In(1+8),
an allowable value for e.
then using log expansions:
ph <n}
for
Parts (b), (c), (d) are proven in the same fashion.
A(C
(e)
For all
that for
a> 0, b > 0, e
n
~b)-G(A)(M
-a)
n
is a 5upermartingale so
't proper we have
eG(A)a-Ab
~
{C
~
J
't
~b,M
't
e
A(C -b)-G(A)(M -a)
't
't
sa}
pIc't ~ b,
Now let's minimize the left, letting
M Sa}
T
b
= (l+e:)a, and we have
'"~ (l+e:)-(l+e:)aee:a
pIc't >- (1 +e: ) a, M't "'a}
~
= ee:a-(l+e:)a
log(l+e:)
Now
2
(l+e:) log (l+e:)
~
e:
e: + 2(1+e:) ,
-9-
so that
ae:
pic ~
(l+e:)a, M
T
T
~ a} ~
2
e.... 2(1+£)
(f) is proven in the same way as (e), using the supermartingale
e
F(A)(M -b)-A(C -a)
n
n
Corollary (1.3).
Assume
Moo
=
Coo = 00 a.e.
, M
11m n
n-+<x> C
n
Proof:
e>0
We show for all
1
a.e.
that
pi:: '
Ie
=
Then
1 + e i.
o.}
= 0 •
Now the probability above is the same as
lim p{Mn
t-+<x>
C
n
~1
+
e'
some
C
n
~ t} ~ lim
t-+<Xl
e
= 0 .
In the same manner
Q.E.D.
-10-
(II)
Martingale Difference Sequences
We assume throughout that
and
E{fn IBn~ I}
= 0,
{f , B}
n
n
is adapted with
If n I :5 1
We define
n
S
n
= L fk
V
= E{fn2 IBn- I}
T
n
=L
v
1 k
n
1
2
n
n
L =
f2
n
1 k
L
In all that follows we assume that
a.e,
T00 =L00 =00
e.
Let
K().) =e
A
-I-A
H(A) = e-A - 1 + A
Theorem (2.1).
{Freedman [ ]}
~ = e
R
n
Proof:
It
E{fIA}
= 0,
A~ 0 •
= e
ASn -KCA)Tn
is a supermartingale
AS -H(A)T
n
n
is a submartingale .
suffices that for
one has
f
with
Ifl
:5
1 and for a a-field
A with
-11-
(a)
and
(b)
To establish (a), it suffices that
e
AX
This inequality obtains for
~
X
I + Ax + K(A)X
= O.
that the derivative on the left in
on the right in
2
To extend to all
x,
it suffices
A is dominated by the derivative
A·, this follows from inequalities established in
Theorem (1. I) .
The proof of (b) is more involved; we sketch a variant of Freedman's
.e
Assuming regular conditionals, replacing
method.
P{o\A}
by
P,
we
show that
We first assume that for
b >0
P{f=b}=p,
where
p + q = I, pb - q = O.
P{f=-I}=q
Some algebra then determines that i t suf-
fices that
I +be-(l+b)A ~ (l+b)e(e
This obtains for
derivatives in
A= 0
A.
and then for all
Next, let
0
$ C $
1, b
-A
-l)b
A from an examination of the
~
0
and assume
-12-
= P , P{f;:: ... cl = q
P{f =b}
p+q
pb ... qc
1 ,
;::
Then
I eAfdP
(AC)
;::
J
-f
c dP
e
o .
;::
~
H(A C) If 2dP
2
e c
2
H(A)If dP
~
e
~
2
c H(>..)
which is once more established by examining the case
A =O.
where this last inequality is obtained from
derivatives and using Theorem (1.1).
Il(Ac)
The next cast involves
for
cs1
taking
f
where
e.
where
1 ~ak ~O, 1 ~bk ~O, ~Pkak • ~qkbk ;:: 0
can be shown [see Breiman [
~Pk + Lqk
and
]] that any such
f
= 1.
can have its values
and associated probabilities represented in the following manner
~
P{f
=
p{f;::
;k}
=
-bk }
;::
Pk
k sN
qk
k sN
~
where
~
~
Pkak - qkbk
~
;::
0
We then have, suppressing the tildas:
It
• all k s N
-13-
JeAfdP =
~(Pk +qk)E{cAflf;::ak \if= "b k }
~ L(Pk + qk)e
e
~
H(~)E{;f2If=~ Vf= ..bk }
H(A) L(Pk+Qk)E{f2If=ak vf=-b k }
x
e .
the last inequality following from the convexity of
involves arbitrary measurable
f
increasing sequence of finite
o-fie1ds
where
A00 = vA.
n
-1
~
f
~
1.
such that
Let
f
{A}
n
is
be an
A00 -measurable
Then
l'
= n-+oo
1m
~
Theorem (2.2).
where
The final case
Let
T
J AE{fIA }
e
n dP
l'
1m e
n-+oo
2
H(A) JE {f\A }dP
n
be any proper stopping time for which either
T
~
t
for
n~ T
s
~
t
for
n~T
n
or
n
some fixed positive
t.
=
Then
Q.E.D.
-14-
Proof:
{~}
(a) follows from the fact that
(b) will follow £rom the fact that
is a supermartingale,
is regular, i,e.,
T
and
o,
When
nST
=> T s t,
n
we argue as follows
<
S e(K(A)-H(A))t
00
and
JR
n
h>n}
=
Je
A5n
-~K(2A)Tn
T
(~K(2A)-H(A))T
n
h>n}
S e(~K(2A)-H(A))t
thus,
• e
is regular.
Iph >n}
The regularity of
when
T
obvious.
Q.E.D.
Theorem (2.3).
Let
T,S,P
{Freedman [ .. ] modified}
be proper stopping times where
(1)
S ~ P and
(2)
T ~
where
is
a < b
suPS 5 ..5
psns
n
p and T - T
are positive numbers.
T
P
p
€
€
[a,b]
[a, b] .
-15-
Let
H-1 , K-1
eH(A)a
(A)
e
(8)
Proof:
be the inverses o£
_H-l(A)b
H, K respectively.
{ A(S 't -5)
P
~
E e
~
E e
18 }.
P
{ .-ACTS-T)
P 18
~
e
K(A)b
-1
}
~ e -K (A)a
p
Follows immediately from the last theorem when
general case, replace
Remarks (2.1).
f
with
n
f
p+n'
8
n
Then
with
8
p - O.
p+n
For the
Q.E.D.
Theorem (2.3A) may be replaced by
valid for all
A,
whenever
't
is a stopping time depending only on the
for the negative values for S may be treated as positive
't
values by interchanging -f
for f. This observation will be used in
n
n
Section (III).
Corollary (2.1).
Let
't,S,p
(1)
S~p
(2)
't
~
be stopping times where
and
p
and
sup
< < S -S E [a, b]
p-n-j.J
n P
Q
T't - Tp E [a,b]
Then
(A)
e
-
(~ 2A+ A2 + A) b
~
~
e
-/ITa
-16-
e
It is easily seen that
Proof:
),,2
,,2
~
2 (1+5\)
H(A)
2
A
2
and
:5 - -
2
-~
K(")
For part B, invert the monotone functions
Corollary (2.2).
Let
Q.E.D.
2(1+A)
{Freedman [;]}
p be a proper stopping time.
n~ p
sup
=
T
3
Let
T ~
T ... T
n
~
p
P
be defined with
b
Then
p{ sup
(A)
p~n~T
S
n
_S
p
~ a IBp}
e
:5
2(a+b)
and
p{ sup
p~n~T
Proof:
(2)
Isn -S p I ~alB p} ~
It suffices to consider
p
=0
.. a
2
2e 2 (a+b) .
as in the last theorem.
follows from (1) by the substitution of
-f
k
Further
Thus, it is
for
sufficient to show that
p{sup S
n:5T
where
with
T
S
n
= sup n
~
a,
3
T :::; b.
n
n
~ a}
-a
2
:5 e 2 (a+b)
From the last theorem, with
T*
= first
n:5 T
we have
:5 e
p{sup Sn
n:5T
K(A)b
~ a}
from which
:5 eK(A)b-A(a) .
The result now follows by minimization of the right side in
A.
Q.E.D.
-17-
Corollary (2,3),
p{s~p
Proof:
We prove the case
symmetry.
Sn
ow} · +~f
p{su P STl
·n
Sn =
=oo}.
ow}
= ! .
the other case following from
Define
o = sup
T
B = To 1\
Since
B is proper and since
Tl
T
T
~b
S
~
a
~
a+l.
Tl
Tl
1
sup Sn
n:5B
we can use Corollary
(2.1) for the inequality
.e
2
e-(j2)o,+A +A) • Ca+!) s f:AT B
~ P{TB<b}+e-Abp{TB~b}:5
Now let
b
+
00.
Remarks (2.2).
and then set
A
o.
P{TB<b} + e
-Ab
Q.E.D.
This last result allows us to define proper stopping times
T
= inf Tl
S
Tl
~
a .
These stopping times play an important role in Freedman [,]; in particular.
Freedman's proof of the law of the iterated logarithm involves a complex
sequence of computations employing such stopping times - see his Corollary
1.10 and its proof.
Unfortunately [ haven't been able to simplify his proof,
I'll merely quote his result as my next corollary and borrow on it later in
-18-
Section IV.
It is interesting, however, that a fairly simple computa-
tion yields an inequality which "almost" gives the upper half of the
iterated logarithm; this inequality will be presented with proof as
Corollary (2.5).
In Section IV I'll sketch a proof of the "almost"
LIL that it produces.
Corollary (2.4).
G1• ven
for
T
{Freedman [ ]}
2
~ < ~
1
ex < '3' b
defined by
T
a
2
16
(64)'
then, for
ex
ex
-T :s;b,
we have
b > 2" log 2" '
9'
= last
n~p;)T
n
p
p proper, and
2
-~(1+4a)~
1
b
S -S ~alB } >- -2 e
n p
p
Corollary (2.5).
Let
p be proper, let
T
Then, for all
ex
with
=
last
n
~
1
p
a < 4"' all
'J
T
n
2
Proof:
We consider' only the case
3
1 ~< 1
a ' b2 - 25
a2
-2(l+a)1
b
-e
5
~~
b
{ sup
P p:s;n:S;T Sn -S p ~alBp}
~
p - O.
We'll use Corollary ((2.1) B).
Define
T
= inf
n ;) S
n
.
-T p :s;b
~
Then
e - ( V2A
~)
+ 2A(a+l) :s; Je"ATT
a .
e.
-19-
so that
- c
-Ah
2
Let
,-- 2a 8 2 , 8
2
t emporar1·1 y unspec1. f·1e d , except .f or t h e requ1rement
.
A
b
8 ~ 1.
that
We have, since
8 ~ 1 and
2
a ~ 1:
b
- e
Let
8
Keeping
.e
= 1 + ex;
a
3
we have
1
"2:S; 25'
b
a
2
b
~ !.
ex '
and we have our result.
the bracketed expression dominates
1
1
1
- - - >e
2 - 5
e
Q.E.D .
-20-
III.
Invariance Results
For each
a> 0,
let
Ya(t)
= as T
(~~ - TT)
------ f
V2
HI
+ ex
T+I
where
T
= sup
n :;} T
n
~
t
2
a
and let
{
X (t)
a
= as a
La -
- a
t1
a 2J
f2
a
f
a
where
= inf n
a
Note that
2
(a L ,as )
n
n
X and
and
Theorem (3.1).
J
-1:
n ~ a
L
Yare the linear interpolations of the sets of points
2
(a T ,as )
n
Y
a
d
+
e.
respectively.
n
X
a
W and
d
+
W as
a
0
+
where
W is Brownian
Motion:
Proof:
and that
We must show that for all
{X, a> O}
a
O<t
« tN'
·
- I
d
(Ya(t l ),
,Ya(t N))
+
(Xa(t l ) ,
,Xa(t N))
+
and
{Y, a> O}
a
(W t '
,W
(W t '
,W
I
d
I
are tight.
t
)
N
t
)
N
It is clear that the
finite-dimensional convergence will follow if
e•
-21-
(qS
L
,
,qS
l
d
L
) + (W
N
t
,
,W
'
,W
l
t
)
N
t
)
N
and
(qS
n1
,asn )
,
d
+
(W t
N
1
where
t
k
= sup n
C)
Ok
= inf n
3
T
k
T s
n
ex
t
L
n
2
k
~
2
a
We have for all
as
a
+
0 .
This follows from the estimates given in Theorem (2.3A):
1(t k -t2k _1
e
N
H ( a}).
k J)
Letting
a
+
a
0,
the extreme left and right above converge to
~[~ej]2(tk-tk_l)
e
so that
Next, to show
(aS
n1 , ,asnN)
d
+
(W
t1
'
,W
t
)
N
and for
T
= sup n
3
T
n
t
S"2"
a
it suffices that, for any
t > 0,
-22-
~
t
2
ex
we have
ex Is
T
- s0- I & 0
as
ex
O.
-+
We require some estimates.
ex
2
~
2
~
p:::: T v 0.
Then, for all
a
~ ~,
i 8t:
from Corollary (1. 2b).
ex
Let
In the same manner one can show that for
i at:
from Corollary (1.2b).
Combining these estimates with Corollary (2.28):
p{IS p -ST I ~~}
ex ~
P{T p
~(1+2a)T T}
+ P{T p - TT
8
~
e
and the same procedure applied to
2
ex
E
t
-32
ex
p{
~ 28 -.!-2 ' Is P - s T I ~ exE}
+ 2e
2
2 (Eex+28t)
Is po-ex
- s I ~~.}
leads to the estimate
-23-
p{ I$
valid when
£
Letting
T
- $
q
> O.
e~~
et
0,
+
6
I ~~}
~
et
and
et
2
2
t
£
2
2e - -3 et-
~~
4e
+
2
2 (£et+26t)
6t.
we have
p{ I
lim
$
et+0
~}
I
a .::. et
....
- $
T
£
<
-
inf
6
4e
2
- 46t =
To complete the theorem we must prove tightness for
o .
{X}
a.
and
{y},
a.
We
use the criterion given in Billingsley [1], (Theorem 8.3):
Given
(*)
£ > 0, h > 0,
there exists
° >0
and
0
et > 0
o
s~p p{tS$s::+° IZet (S) -Za. (t) I ~ £} ~
.e
Z is
X or
Y.
Note that the criterion above is specified in
Billingsley for processes which live on
on
C[O,oo).
C[O,I]
while our processes live
Criterion (*) remains adequate for
We begin with
00h
0
a.~a.O
where
such that
Y,
T
defining, for
= sup
~
T
n
:5
~
a
2
et
C[O,oo) - see Whitt ['].
0 < t < s:
= sup
n
Tn
~
S
2
et
and reducing (*) to the equivalent form
>£} ~ hOO
sup { sup Is -$ I
t~S~t+oo
a T - et
t
.
a.~eto
Using Corollary (2.2B), we estimate the left side with
.£
p{t:5$:5t+o
sup
o
1$a -S T I >- £1
B }
a. T
< 2e
-
2
-24-
when
£
< 1 and
a
o = 00 < 1 and 00 is small enough.
Now let's consider tightness for
T
= first n
;)
p
= first n
;)
X,
Define
L ~-!...
n a2
L
~2-
n a2
S>t
It suffices that
p{
sup
~ Is p-sTI ~a£1 BT} ~
t~sst+uO
By the usual time translation, sending
p{ISn I ~£a
(**)
f
n
0 h .
0
into
f
it suffices that
T+n'
some
The left side is bounded by the sum of the two terms
p{ Sn
1
I
~~.
some
,
T
n
~ (1+28) [O~
a
+
In
and
{ n a°0
P L
~2+l
,
some
Tn' (l+2e) l~ + I]} .
Using Corollary (1.2C) and Corollary (2,2B), we see that the left side of
( **) has bound
upon taking
£
<1
and
a
= 00 < 1.
O
£
Letting
8
£
= 16'
we have the bound
2
3e - 1200'S 00h
for small
°0 .
Q.E.D.
-25-
Corollary (3.1),
Proof:
Let
a. \10,
Let
-+- 1,
m
On a set of measure one, for each
L
T
m
~
L
n
<
n,
T
m
= first
there exists
n
:l
L
n~
1
2
a.
m
m with
L
T
m+1
Sf:
Then
Tm
__
Tm
+~
-L
n
,JT mm
Now
,e
so it suffices that we show
p
a.m+ lisn -S T 1-+-0.
m
As usual:
p{ISn-STmI ~~}
: : Jp{IS -ST I ~a~IBT }
m+1
n m
m+1
m
and for large
m the 'imtegral wi 11 be bounded by any universal bound for
p{IS I ~_e:_
n
a.m+ 1
,
some
L
n
~+22}
a.
a.
m+l
.
m
Now use Corollary (1.2) and Corollary (2,1) for appropriate
m-+-oo ,
e
and let
Q.E.D.
-26-
(IV)
Iterated Logarithms
Theorem (4.1).
p{S
(2)
i,
e~
For all
(1)
n ~ (1+28) I2Tn
e
For all
all
log log
with
N
~
N
8
2
""4' we have
1
some
log log (T) ,
n
(1og (N)) e
log log N
N
~
1
4' all N with
~
(N)
3
16
2
8 ,
1
some
e
In particular, for all
we have
(log N)
e
> 0:
(3)
where
M =T
n
n
Proof:
Fix
0< t
or
o
e.
L .
n
<t
and define
1
TO
= first n
Tl
= last n
;I
T
{first
=
n
;I
T
T
l
T
::>
n
n
if
~
to
t
~
l
[TO,T l ]
no such
ep (T )
S
~
n
occurs
n
n
where
ep(t)
=
(1+28)12 t log log (t) .
Then
1
I
~
{S
~ep(T
T
e
)}
T
AS -K(A)T
T
T
~
I
{ST~ep(TT)}
e
Aep(T ) -KCA)T
T
T
-27-
and this implies
where
The minimum over
x
x - T
1"f
t
1
10g(l+x)
~
2
for
x~1
8 ~ '3'
1
an d
= (1+8)m+1,
we have
= (1+8)t O. Repeating this argument for
1
and denoting the set under consideration as
t
~
P{A }
m
8 ~
t
t
o=
A,
m
(1+8)m,
we have
1
1
1+8
(in(1+8))1+8
m
2
m
8
log log (1+8) ~ ~
It is easily seen that the
(l+8)m
last constraint holds for all m ~ m provided it holds for mO. Thus,
O
under such circumstances:
provided
and
1
1
(in(1+8))1+8 •
Since
mO ~ 2
p{Sn
Letting
~
~<P(Tn)
N =
m -1
O
~
m 10g(1+8),
O
, some Tn
m
(1+8) 0 with
we have
~ (1+8) mo}
log log (N)
N
e·
1
~
1
(in(l+8)) 1+28
2
8
-< - 4 ,
we have
-28-
p{S 2: <P (T ) ,
n
some
n
Tn 2:
N}
-01
'$
•
1
~
e.
Next, letting
£
~
41
and
p{:n
t [1-£,1+£] ,
some
1
1
(tnN)8
1
7/3
some
L 2: N}
n
L
n
n
+ p{Sn 2: (1+2M8)12(1-£)T
n
2:N}
log log (l-£)T
n
,
some
Tn 2: (l+£)N} .
We now use Corollary (1.2b) and (1.2c) and the fact that if
an d
1
th en
£ -< 4'
of
1
8
M temporarily unspecified:
p{Sn 2: (1 +2Me) 12L log log L ,
n
n
~
8
(tnN)
(i~-(-~+8))4T3
< -1
_.
-
1
1
(bl (l +0)) 1+6
log log (l-E)T
~
(1-£) log log T
log T 2: 3.2
so that
2
-
2e
~ N
8
and
p{Sn 2: (1+2M8)/2(1-£)T
~
for
log log (1-£)T
p{Sn 2: (1+2M8) (1-£) 12T
£ = e, M = 2, 8
p{Sn
~
-
41
e
7/3
n
n
,
log log Tn'
some
some
Tn 2: (1-£)N}
Tn
~ (1-£) N}
we have the last term dominated by
~ (l+Z8)/ZTn
< _2_
true if
n
log log (Tn) ,
1
(1og(1-e)N)
if
8
some
Tn 2: (1-e)N}
2
log log (1-0) N
8
(1-8)N
~ 41
'
-29-
log log N < 3
N
-
16
8
2
Thus,
p{sn ~ (l +48) /zLn
2
8
s;
2e
s;
2e
- TN
1
8 7/ 3 (10g(1-0)N)0
21131
+ -- ---
For all
Q.E.D.
8 > 0:
We prove the result for
Freedman's proof for
log (M)
n
n = Ln,
M
i.o.}
= first
M =T .
n n
n
3
L ~ (l+£)r
n
k
,
r > 1 .
where
~(t)
and where
r, e:
= 12t
= 1
modelling the proof on
Let
T
1
-----=-
N.
p{sn ~ (l-8)/zMn log
Proof:
< --
0 7/ 3 (1_8)8 (log (N))8 - 8 7/ 3 (log N)8
under the restrictions on
Theorem (4.2).
2
+ - - -- - - - - . " .
log log (t)
are chosen so that for a given
0 > 0:
-30-
We estimate
P{AkIBL}
by finding a universal bound for
k
P{Sn
~
1)
k+l ) ,
( 1 - /; cj> (r
For arbitrary positive
p{Sn
Assuming
a
~a,
=
a,b,
some
Tn
some
L
n
k}
(l+£)(r k+l -r)
$
we have
~b} ~
k 1
(1 _Jr)cj>(r + ), b
P{L
n
~ (l+£)b,
+ p{Sn
~a,
= r k +1
- r
some
some
k
L
n
~b}
Tn
~ (1+£)b}
and that
is large enough
k
for Corollary (2.4) to hold, and using Theorem (1.2), we have
k
P{A IB }
k Lk
~
1
2(10g
thus we see that for all
r (r-l)£
2 (l +£)
1
_..::---::;- e
(k+ 1) e
r)8
£ > 0,
2
all
r > 1, 0 > 0
for which
e
< 1,
we
have
Using Corollary (1.1) with
for infinitely many
gn
-S
n
~
thus for fixed
£ > 0 ,
'
we see that
n
k:
Now Theorem (4.1), with
= IA
in place of
=
rIA
00
a.e.
so that,
k
Sn
k
demands that, eventually
- (l+e:) cj>( (1+£) r + 1)
e,
-31-
eventua11Y1 consequently, for infinitely many
k;
from which
Sn
~
- (1+2£)
I:r
<P (r
A,l <P (rk +1)
k +1) + (1 _
vrJ
n:
from which, for infinitely many
S
n
~
,
l(l - (2+2£))<P((1_£)L )
rr
for small
n
, thus for large r,
E'
small
£
Q.E.D.
Remarks (4.1).
Theorem (4.2) depends strongly on the inequality given in
Corollary (2.4).
As mentioned earlier, the simpler result Corollary (2.5)
almost yields the L.I.L.
That is, using Corollary (2.5) and imitating the
proof of Theorem (4.2), it is easy to show that
p{Sn'
for
•
n = Tn
M
or
Ln'
(1-e)
JM!)
log log
Mn
i. o.} =
1
-32-
Remarks on First Passage Times
Suppose
{f ,8 }
n
n
E{f
Fix
to
~
n
is no longer centered but satisfies instead
18n- I} ~
aE{f
2
18 I}'
n n-
some
a
E
(0,1]
12 to log log to if t
~
to
~
to
and define
6
lfl(t) =
1(1+8)
l
(1+8) 12 t log log t
if
t
We want to consider
T
where
[].
8
~
t·
=
first
We will consider
The time
T
n
I'r
3
S
T
n
~
lfl(T )
n
in the
spirit of Darling-Robbins
is proper; we define
gn
= !2
(f -E{f IB
})
n
n n-l
Using the second part of (A) we have
Sn ~ (1-8) 12 Tn log log Tn
i.o.
S
Then using the first part of (A) we conclude
follows that
T
is proper.
Thus, with
""
-S
n
lim Tn ~ a,
n n
in place of
from which it
we have
•
-33-
I
I
~
e
1.(-5 )-K(A)T
T
T
{T ~t}
T
We note that
log log t
is decreasing for
t
t
~
6,
so if we assume
t
~
to
~
6
and if we further assume
O
(C)
~ K(A) < 2
A
_ 2 (<j>(t)+l)
at'
-
then
A straightforward calculation shows that the right side minimizes for
__
K' (A)
2a -2
(<j>(t)+l)
t
or
A=
.e.n.{ I
+
2a _ 2 ( <P ( t ) + I)
t
Since
1
f
this value satisfies the constraint (C) and is there-
K(A) ~ AK'(A),
fore available.
2
Using
x
2(1+x) ,
(l+x)log(l+x)
t
P{T
T
~
Now our assumption that
in
t
t
t} ~ e
t
~
to
2(l+2a)
~ 6
with a maximal initial value
for which
all
~x+--­
x
~
0,
we have
(a _ (<P (t{ + I) '\ 2
)
implies that
<P(tO)+1
to
<P(t)+l
t
Choose
is decreasing
t1
as the unique
-34-
some
b < 1 where
Then
Let
2
a
(1_b)2 t
2 (l+2a)
c =
(I-b) 2
~".--;;,;'--=­
2(1+2a)'
we have
S t
The defining property of
+
l
t
l
1
-ca
--2- e
ca
2
t
1
S t
l
1
+ ---2
ca
demands that
Le., that
log log
t
1
A crude bound is
yielding
JTT
For example, when
S
to
2
2
c
e
6 { 1
-e + -log
-log
2
b
a 2 (1_b)2 + b 2
b
=6
and when
e = 13
we have
i} .
-35-
so we can take
b
= .75
and we have
Note that the method used in finding such upper bounds will work for
all
as
for which
t -+
00
•
¢ (t)
t
is eventually decreasing and has limit zero
-36-
References
.
Convergence of Probability Measures.
[1]
Billingsley, P. (1968).
Wiley and Sons.
[2]
Breiman, L. (1968).
[3]
Darling, D.A. and Robbins, H. (1967). Iterated Logarithm Inequalities. Proc. Nat. Acad. ,';ci., Vol. 57, 1188-1192.
[4]
Drogin, R. (1972). An invariance principle for martingales.
of Math. Stat., Vol. 43, 112, 602-620.
[5]
Freedman, D.A. (1973). Another note on the Borel-Cantelli lemma and
the strong law, with the Poisson approximation as a by-product.
Annals of Probability, Vol. 1, #6, 910-925.
[6]
Freedman, D.A. (1975). On tail probabilities for martingales.
Annals of Proability, Vol. 3, #1, 100-118.
[7]
McLeish, D.L. (1974). Dependent central limit theorems and invariance
principles. Annals of Probability, Vol. 2, #4, 620-628.
[8]
Rootzen, H. (1977). On the functional central limit theorem for
martingales, Zeit. Wahr., Vol. 38, #3, 199-210.
[9]
Whitt, W. (1970). Weak convergence of probability measures on the
function space C[O,oo). Annals of Math. Stat., Vol. 41, #3,
939-944 .
Probability.
John
Addison-Wesley.
Annals