PROCEEDINGS
THE
PROCEEDINGS OF THE
AMERICAN
MATHEMATICAL SOCIETY
SOCIETY
AMERICAN MATHEMATICAL
Volume 89, Number
Number 4, December
December 1983
Volume
ALMOST SURE STABILITY OF PARTIAL SUMS
SUMS
OF UNIFORMLY BOUNDED
BOUNDED RANDOM VARIABLES
VARIABLES
l
THEODORE
THEODORE P. HILL
HILL1
ABSTRACT.
al, a2
a2,...
sequence of real
oo. If
ABSTRACT. Suppose aI'
•... is a sequence
real numbers
numbers with a/1
a,, ~ 00.
= a a.s.
limsup(
+ ... + X,
for every
every sequence
sequence of independent nonnegative
a.s. for
lim
XI +
X/1)/a/1
sup( Xl
)/a,, =
uniformly bounded random
variables Xi,
uniformly
random variables
XI' X2,...
X 2,. .. satisfying some hypothesis condition A, then
for every
every (arbitrarily-dependent)
(arbitrarily-dependent) sequence
sequence of nonnegative uniformly
uniformly
then for
= a a.s.
bounded random
random variables
/ a /1 =
the set
variables Y
Y1,
Y,I + ... + Y:,)
Y, )/a,,
a.s. on the
sup( Y
Y2,...,
I, Y
2 , .•. , lim sup(
where the
distributions (given
(given the past) satisfy precisely
where
the conditional distributions
precisely the same
then the assumption
condition A. If, in addition,
addition, ~oc;a~2
Y.a-2 < 00,
assumption of nonnegativity may
oo, then
be dropped.
dropped.
1. Introduction.
Introduction. The purpose
answers to the
1.
purpose of this note is to provide partial
partial answers
question of when the following statement
true in the case where"
property B"
statement (81)
(SI) is true
where "property
B"
behavior of the stabilized (or normalized)
normalized) partial
is a property
property reflecting
reflecting the limiting behavior
partial
sums.
sums.
(81)
(S1)
If every
random variables
every sequence
sequence of independent
independent random
variables having
property A has property
property B almost surely,
property
surely, then every
every sequence
of random
random variables
variables has property
property B almost surely
surely on the set
where
where the conditional distributions
distributions (given the past) have
property A.
property
true for
for all choices of properties
properties A and B,
Of course
course (81)
(SI) is not true
B, but it is sometimes
true in rather
rather general
for example, it was shown that
that (81)
true
true
general contexts. In [4],
[4], for
(SI) is true
whenever"
property B"
partial sums
property A. In
B" is "the partial
sums converge",
whenever "property
converge", regardless
regardlessof property
present note, a partial
partial analysis of (8
I) is given for
for conclusions concerning the
the present
(SI)
stabilized partial
limiting behavior,
behavior, not of the partial
partial sums Sn'
Sn, but of the stabilized
partial sums Sn/an
Snlan
for suitable
suitable normalizing
normalizing constants {an}'
{a) }.
= (Y
random variables
As in [4],
[4], 61J
6J ==
(Y1,
Y2,...)
variables on a probability
I, Y
2 , • •• ) is a sequence of random
=
+
--+
Y1
the
triple (D,
~, P), Sn
==
Y
+
...
+
Y
,
and
~
is
sigma
generated
field
triple
(2, A,
generated by Y
Y1,...,
Sn
Yn,
Fn
Yn.
n
I
I ,· .. , Y
n.
H ==
for Y
distribution for
Let 'TTn(-,)
regular conditional distribution
~n-I', and let IT
Yn
n given 6nY
n(', .) be a regular
('TTl'
product a-field on Roo;
Ji denote the Borel
Borel a-field on R,
Ji? the product
R, and g?Joo
R?;
(r 1 'TT
7T2).
2 , • •• ). Let g?J
let ~(R)
== @(R)
~(R) X
measures on (R, g?J),
C,=
@(R) denote the space of probability
probability measures
J33),and let e
- . (As mentioned in [4],
~(R)
reader to think
think of 61J
random
@(R) X ....
[4], it might help the reader
6J as a random
Received
June 10,
Received by the
the editors June
revised form,
10, 1982
1982 and,
and, in revised
form, April
April 4, 1983.
1983.
1980
Mathematics Subject
15; Secondary
1980 Mathematics
Subject Classification.
Classification. Primary
Primary 60F
60F15;
Secondary 60G45.
Key
words and phrases.
Key words
phrases. Almost-sure
partial sums,
numbers,
Almost-sure stability of partial
sums, martingale
martingale strong
strong law of large
large numbers,
conditional generalizations
laws.
generalizations of strong
strong laws.
I' Partially
Partially supported
supported by a NATO Postdoctoral
Postdoctoral Fellowship.
? 1983
Mathematical Society
Society
©
1983 American
American Mathematical
+ $.25 per page
0002-9939/83 $1.00 +
0002-9939/83
685
685
686
T. P.
P. HILL
HILL
T.
element of Roo,
rI as a random
X) is the distribution
RI, and of IT
random element of e.)
(C.)e(
distribution of the
E(X)
random
X.
random variable
variable X.
formalization of statement (81)
(SI) is the following
Let B E
E ~oo.
q?'. In this notation, the formalization
statement
statement (82)
(S2) of [4].
[4].
C is such that (XI'
If AC
a.s. whenever
ACe
X 2 , • •• ) E B a.s.
whenever Xl,
XI'
(X,, X2,...)
are independent and (e(X
(82)
X 2 , ••• are
e(X2),...)
for
(S2)
X2,...
I ), e(X
(P-(X,),
2 ), ••• ) E A, then for
H E
arbitrary
61J, 6?
61J E B a.s.
E A.
arbitrary 6,
a.s. on the set where
where IT
oo (S2) is true
DEFINITION
E 0D
ACe}.
1. X:=
C &}.
true for all A
DEFINITION 1.
XC= {B E
qJ?:: (82)
Throughout this note, aI'
and b,,
b2,...
are sequences of real
real numbers
numbers with
... and
Throughout
bl, b
a,, aa2'2 , •••
2 , ••• are
=
r<
-x 00
o and
and r:= (r
an
and 2.oob,~2
result in this paper
0o,00, and
a,, ~
E'b,;2 <
(r,,l , rr2,...).
2, .. .). The main positive result
is the following theorem.
theorem.
< 00,
THEOREM 1. For
THEOREM
For all
M > 0, all-oo
allM>
all -x <
<aa <
<bb <
C [-00,00],
xo, and all D C
[-oo, xc],
B Btt
< rr,1 ?~ Mfor
{i R?cz
+ ...- +rn)/a
=B
(r l +
(i) B:=
{rE
ROO:: 0 ~
M for all i, and
lim sUPn-+oo(r
+r,)/a,, n E
andlimsup,,
D} EX;
eSC;
and
? b for all i, and limsuPn-+oo(r
B =B B;
(ii) B:=
R?: a ~
s rr,1 ~
+r,,)/bnn E
{Bab {rE Roo:
limsup,, (r,l + ...- +rn)/b
D} E
1(.
LSC.
Moreover, the
analogs of (i) and (ii) with
hold.
Moreover,
the analogs
with lim sup replaced
replacedby
by lim inf or
or lim also hold.
°
In contrast to the conclusions of Theorem
Theorem 1,
1, examples will be given in §4
?4 to show
that
that
(1)
{r
={ (r E
B=
Roo: -1
,,;;;; r, ,,;;;;
for all i, and
and
ER:
-I?r,
s 1I foralli,
lim sup (r l +
+
11-00 00
11-+
... + rr,n )/(n
) / (n log log(
n/2)) 1/2
log(n/2))1/2
1) (4
;
and that
(2)
B=
=
{rEE R??:
Roo: lim (r
11-+
11-0X00
1
+ ·*-+rn)/n
. . + rn ) / n #:;6 0}
o} El ~.
W.
theorem for
l(i) essentially says that any theorem
Theorem 1(i)
Theorem 1. Theorem
2. Applications
Applications of Theorem
variables which concludes
random variables
nonnegative uniformly
uniformly bounded independent random
"limsuPn-+ooSn/an
a.s." may immediately
"lim sup,, OOSn/anE
E D a.s."
immediately be generalized
generalized into a conditional
fact.
version of that result.
For
example,
suppose
one
just
learned
the following fact.
learned
result.
(3)
variables taking
are independent random
If X1,
XI' X2,...
X 2 " •• are
random variables
taking values
in [0,1], and
x n - I for all x E [0,1] and
< x)
> 1 - xnand if P(X
X) ~
P(X,,n ~
all n, then
< 11 a.s.
+ ...
limsup(X
lim sup( X1
* * +Xn)/logn
+Xn )/log n ~
I +
Applying Theorem
an
Theorem l(i)
1(i) with an
alization of (3) given by (4).
(4)
log n, M
M = 1,
1, and D :=
= [0,1]
[0, 1] allows the genergener-
random variables
variables
If Y
are (arbitrarily-dependent)
Y1,
Y2,...
(arbitrarily-dependent) random
I ,Y
2 , ••• are
a.s. on
taking
sUPn-+oo. Sn/log
taking values in [0,
[0, 1],
Sn/log n ~ 1 a.s.
1], then lim supn
I
n I
the set where
~
xl~_I)
~
1
Xnx
for
all
x
E
[0,1]
<
where P(Y
>
1
E
P(1Yn
n
xI167)n
and all n.
STABILITY OF PARTIAL
PARTIAL SUMS
SUMS
ALMOST
SURE STABILITY
ALMOST SURE
687
Clearly
versions of the
Clearly it is not difficult to prove (4) using the conditional versions
arguments
that
Theorem 1 is that
case; the point of Theorem
arguments used to establish (3), as is often the case;
this is not necessary.
necessary.
martingale strong law
As an application
Theorem I(ii),
l(ii), one may prove Levy's
Levy's martingale
application of Theorem
of large
proving the special case of independence, and
large numbers
numbers [5,
[5, §69,
?69, p. 250]
250] by proving
=
1(ii) with bbn
n and D == {O}.
{0}. For a stronger
==
nand
result,
similar result,
then applying
stronger similar
applying Theorem
Theorem I(ii)
n
consider first
first the following theorem
theorem (5).
(5)
[6,
XI' X2,...
X 2 , ••• be a sequence of indepen[6, Theorem
Theorem 2.8.1].
2.8.1]. Let Xl,
uniformly bounded random
random variables.
dent, mean zero
variables. Then
zero uniformly
l 2 E
> O.
(XI
)/n / + ~
for each
E >
- 0 a.s.
+ ...
a.s. for
each E
0.
(X1 +
*-- +X
+Xn)/nl/2+EO
n
generalizaimmediately has the following martingale
martingale generalizaApplying Theorem
Theorem I(ii)
1(ii) one immediately
tion of (5).
(6)
(6)
martingale with uniformly
S1,S2,...
S2"" be a martingale
uniformly bounded
Let SI'
l 2 E
->
increments.
Then
Sn/n
/
+
~
0
a.s.
for
each
E
> O.
-0.
increments.
for
/2+e
Sn/nl
(Stronger
results than
than (6) are
for example, Chow's
result [2 or 7,
are known;
Chow's result
known; see, for
(Stronger results
Theorem
that the classical formulations
formulations of most laws of
Theorem 3.3.1].)
3.3.1].) It should be noted that
framework of Theorem
Theorem 1;
iterated logarithm
the iterated
1; they usually involve
logarithm do not fit the framework
some form
form of centering
assumption of (i), and
centering which violates the nonnegativity assumption
= (n log log nn)1/'2
clearly
==
)1/2 does not satisfy the ~oob;2
< 00
?b- <
x hypothesis of (ii).
clearly bbn
n
Although the above applications all involve conditional moment
moment hypotheses, the
"hypothesis"set A
A in (S2) need neither
neither be measurable
measurable nor involve (conditional)
moments. (For a similar
reader may see [4,
similar nonmoment application, the reader
Theorem 2].)
moments.
2].)
[4, Theorem
It should
remarked that
that technically
technically speaking
should also be remarked
Theorem 1 may
speaking the set D in Theorem
also be nonmeasurable,
applications, D will simply be a point or
nonmeasurable, although
although in most applications,
interval.
interval.
3. Proof of Theorem
Theorem 1. The argument
argument below closely parallels
parallels that in the proof of
[4,
Theorem 1];
crucial difference being the use of Lemma
Lemma 2 in place of the
[4, Theorem
1]; the crucial
three-series and Borel-Cantelli
Borel-Cantelli arguments
three-series
arguments of [4].
[4]. Without loss of generality,
generality, assume
(D,
(Q, ~,
A, P) is complete.
LEMMA
LEMMA1 [4, LEMMA
LEMMA1].
1]. (S2)
(S3)
(S3)
~
#
(S3).
all 6?, where
where Prr(w)
P(
{w: Prr(w)(B)
f/:. B) ==
= I}
= 0Ofor all6]J,
1) n
P({w:
nfl6]J i4
Pr(,,)(B) ==
Pn(,) is
...
(
(
w)
'1T
w)
...
on
(ROO,
0?>OO).
the
product measure
X
X
the product
measure '1T
on
(R?,
ffi?).
72(w)
r,(w)
I
2
REMARKS. In Lemma
replaced
Lemma 1,
REMARKS.
1, the "hypothesis" set A of (S2) has in effect been replaced
for the conclusion
representing all possible sufficient conditions for
by the set representing
"(XI'
X 2 , ••• ) E B" to hold; one direction
direction of the equivalence uses the completeness
"(X1, X2,...)
of the measure
terms of
all subsets of null sets are
measure space to guarantee
are measurable.
measurable. In terms
guarantee all
Lemma
Lemma 1 says
the class %,
YuC,
= I}
%
== {B
p({w: Prr(w)(
B) ==
= 0 for
(i4B)
n6 6]J f/:.
for all 6]J}
6}. .
1} n
SC,=
{B E
E 0?>oo:
'??o: P({W:
B) ==
Prl(,)(B)
LEMMA
LEMMA
2. Let Y1,
YI , Y
sequence of uniformly
uniformlybounded
bounded(arbitrarily-dependent)
(arbitrarily-dependent)
Y2,...
2 , ••• be a sequence
= YEn
6.
let sn
random
F ) Then
Then
sn ==
variables, and let
random variables,
~7 E( lj
YjII~_I)'
a. s.; and
(i) lim sUPn_oo
Sn/bn ==
lim sUPn_oo
sn/bn'a.s.;
sup
=MSUPn
Sn/bn
oosn/bn
lim SUPnoSnl/an a.s.
Y
lim
all
then
a. s.
(ii) if 1'; 2
~ 0 for
i, then
sUPn_oo
supn n . Sn/an
Sn/a= == limsuPn_oosn/an
(7)
(7)
688
688
T. P.
HILL
P. HILL
T.
PROOF. (i) follows from
from a martingale
result of Chow [2 or 7, Theorem
Theorem 3.3.1].
martingale result
3.3.11.
-x
(ii) follows easily (since an
~
(0)
from
a
generalization
of
a
result
of
from
result
Dubins and
CC)
generalization
an
Freedman [3]
Freedman
Brown [1],
[3] which can be found in Brown
[1], namely:
namely:
If Y
uniformly bounded random
random
are nonnegative uniformly
Yl,Y2,...
1, Y2 , • •• are
=
<
then
variables,
cc
1
a.s., and == almost surely
variables,
lim
n_ 00- Sn/sn
Sn/S n < 00 a.s.,
surely
limn
- 00.
cc. C]D
on the set snsn ~
(8)
The following example shows that the "uniform boundedness"
boundedness" assumption
assumption in
" tightness".
Lemma
replaced by "tightness".
Lemma 2 may not be replaced
EXAMPLE 1. Let Y
EXAMPLE
with
independent with
Y2,...
Y,,
1, Y
2 , ••• be independent
2
2
= 0).
= nnn2)
) == n- 12=
p(Yn ==
==
p{Y
= 1 -- p(y
)=n
(n n ==
=?
P(Yn
sn/n == 1,
a.s. by the Borel-Cantelli
Thensn/n
Borel-Cantelli Lemma.
Lemma.
1, but lim sUPn_oo
Then
Sn/n
supn
Sn,/n == 0 a.s.
PROOF OF
for
(i)
only
will
be
given;
PROOF
OF THEOREM
THEOREM1. The argument
for
for (ii) and
argument
given; that for
lim inf or lim are
are similar.
similar.
C [M > 0 and D C
00, 00].
Fix M
[-cc,
cc]. By (7), it suffices to show that B satisfies (83),
(S3),
= {rE
{Fr Roo:
= Bo
.
M for
where
+
R?: 0 <~'1r, <~ M
where B ==
BaM ==
for all i, and limsuPn_oo('1
+
...
+'n)/a
(r1
limsupnr.)Ian
n
ED}.
E
D}. Fix 6]J
6J == (Y
)
Y2,.
2 , •• • ).
(Yi,1, Y
and (enlarging
~, P)
~,P),
Let (D,
(Q, A,
P) be a copy of (D,
(Q, A,
P), and
(enlarging this new space if necessary)
necessary)
w), Z2(W),...
Z2( w), . .. be a sequence of independent random
random variables
variables
for each wED,
X E Q, let Zl(
Z1(w),
=
C [0,
of
on (D,
~, P)
==
'17
Let
==
{w
E
D:
support
'17
(w)
C
(w).
E
{w
E
(Q, 9X,
P) with e(Zn(w))
Q:
support
[0, M]
Tn(w).
7n(w)
n
(4Z,n(w)) n
that, without loss of generality,
and observe
assumed that
for all n}, and
observe that,
generality, it may be assumed
> I}.
~ Z,z<
w) <
~M
M everywhere
for all nn ~
2: 0 <
{w E
e D:
E == {w
everywhere (in D)
2) for
1). Next, calculate
Zn,(w)
(9)
(9)
=1}I}
n 6]J fl B)
iI n
P((w:
p( {w: Prl(,)(B)
PTI(w){B) ==
(i B)
n6 6]J fl
==
P(E n
PTI(w){B) == I}
B)
{w: PHI(W)(B)
1) n
n {w:
=P(E
Z;(
= P ( { w:
< Zj(w)
< M
=
w) :;;;;
M for
for all ii and
and
0: 0O:;;;;
4 ff.
i B)
limsup(Z1(w)
(s2, if,
X, ]5))}
PS)) n ~
6@
li~S;P(ZI(w) + ... +Zn(w))/a
+Zn(w))/an n E D a.s. (in (D,
ED
,, -
{w:
B)
00
w):; ;
= P ( w: 0O :;;;; Z,
< M
M for all ii and
and
=
(
Z,(w)
lim sup (EZ1(w) + ...
E D}
li~s;P(EZI(w)
* . +EZn(w))/a
+EZn(w))/an n ED}
,, f
00
n
B)
~ ff.i4 B)
= P(E
w: limsup
(EY1 + E(Y2I'
=
p( Enn {w:
li~s;P(EYI
E(Y2 Y1)(w)
'?J])(w)
1
,, -
00
+
+**... +E(Y
+E(Ynn II'!f,,-])(w))/a
(gn-1)(w))1ann
ED}
B)
n ~ ff. B)
D} n
E
6@Mi
==
-0, 0,
first equality
where the first
equality in (9) follows by the definitions of E and B;
B; the second by
Lemma
Zn(w)
and
B;
third
by
Lemma
2(ii)
(recall
the
third
the definitions of Zn(
Z1(W),
w)
B;
(recall that ZI(W),
Z2( w), ....are
.. are independent in (D,
~, P)
for each w);
fourth by the definition of
(Q, 9X,
P) for
w); the fourth
Z2(W),
and E. D
Zn( w) and
Band
Lemma 2(ii) again
and the last by Lemma
and '17
again and the definitions of B
CI
vn;
Zn(w)
n; and
ALMOST
ALMOSTSURE
SURESTABILITY
STABILITYOF
OFPARTIAL
PARTIALSUMS
SUMS
689
689
4. Examples
Examplesestablishing
and(2).
establishing(1)
(1) and
(2).
4.
=
EXAMPLE
2
Let
[4].
S
= 0)
where SI'
are iid,
EXAMPLE 2 [4]. Let YYn
Sn-I' where
iid, P(Sn
P(Sn ==
S2,'*. are
0)
SI, S2""
n == Sn -- S_,
=
=
=
/2
P(Sn == 1)
1) == 1.
4.Then
Then for
for an
a == (nloglog(n/2»1/2
{ 1}, limsuPn->ooSn/an
(n log log(n/2))' and
D == {I},
== P(Sn
and D
lim supn 0Sn/a
= 0Of/:.M D
D a.s.,
a.s.,but
but PIl(w)(B)
for
a.s. for
==
PI(,)(B) === 11a.s.
{r
- I :so;; r,
FE Roo: -I
BB== {rR?:
r, :so;; IIfor
forall
alli,and
i, and limsup
lim sup (r,
(r, ++
n oo
n-oo
... ++ rrn)lan=
n ) /an = I }).
Via (7),
(7), this
this proves
proves (1).
(1).
Via
= Iii)
EXAMPLE 3.
3. Let
Let SI'
be independent
Sl, S2""
with P(Sn
independent with
)n == P(Sn
EXAMPLE
S2,.'.*.be
= 1
P(Sn ==
P(Sn == -Iii)
2) ==
=
=
- Sn
and let
let YYn
From
the
(Y
definition
of
one
and
==
Sn
Sn-I
(Y
==
SI)'
From
the
definition
of
Sn'
one
has
that
lim
Sn/n
has
that
SI).
lim
Sn
Sn/n
Sn,
1
n
There are
a.s. There
are only
two possible
conditional laws
only two
laws for
for '!Tn'
== 0Oa.s.
possible conditional
namely
vn, namely
8
(+n== (s(
'IT~+)
J; +
ln ) ++ s(
8 (-rn
rn~- )l)/2,
;;;-=1)
-J; ++ ;;;-=1)
)/2,
and
and
7n(-
(rn -
n -1
+ 8 (-n-
rn~- f))/2.
{Z} as
Construct {Zn}
as in the
the proof of Theorem
Theorem 1.
Construct
1. Then
Then the
the unconditional
unconditionaldistribution
distribution of
{ Z1 } is independent with
with law
law
{Zn}
p(
=
P ( Zn
+
Zn = J;
S +
- n 1 ) = P(
;;;-=1)
p( Zn
Fn +
+ ;;;-=1)
lln 1 ) = p(( Zn
=FJ; - ;;;-=1)
Zn =
Zn = -J;
lln 1 )
== p( Zn == -J; - ;;;-=1) == i.
It will now be shown that
(10)
(Z1+
+Zn)/n-1+0
a.s.
To see (10), consider the following theorem of
of Revesz [6, p. 65 or 7, p. 167]:
167]:
(11)
If Z1
ZI'I Z2,...
Z2"" are independent with IIZn I<
I~ n a.s. and E(Zn)
£(Zn) === 0
> 1, then (Z1
for all n ~
-O
+
(ZI + ... +Zn)/n
+Zn)/n ~ 00 a.s.
a.s. implies
implies
22)]/log
=
lim nn
O
.
->
oo[~n£(
Z?
/i
)]/log
==
O.
,:[En
nE(
Zi2li
n
n
Since E(Zn2)
0. nE(Z72/i2)/logn
£(Z;) =
== 2n
2n -- 11 for
for all n, then
then limnlim n ->ooL
£(Z,2/i 2)/log n === 00 which
which by
by
=
(11)
implies
0
(10).
(11) implies (10). Since
Since limnlim n -> ooSn/n
00 Sn/n == 0 a.s.,
a.s., this
this establishes
establishes (2).
(2). (Informally,
(Informally, (2)
(2)
says
says that
that "Sn/n
"Sn/n does
does not
not converge
converge to
to zero"
zero" is
is not
not aa "property
"property B"
B" for
for which
which (SI)
(S 1)
holds
holds for
for every
every property
property A.)
A.)
ACKNOWLEDGEMENTS.
ACKNOWLEDGEMENTS. The
The author
author is
is grateful
grateful to
to Robert
Robert Kertz
Kertz for
for several
several useful
useful
conversations,
to
conversations, to Thomas
Thomas Kurtz
Kurtz and
and the
the referee
referee for
for several
several suggestions,
suggestions, and
and to
to the
the
Department
Department of
of Mathematics
Mathematics at
at the
the University
University of
of Leiden
Leiden for
for its
its hospitality
hospitality and
and
technical
technical assistance
assistance during
during the
the academic
academic year
year 1982-83.
1982-83.
REFERENCES
REFERENCES
1.
1. B.
B. M.
M. Brown,
Brown,AA conditional
conditionalsetting
settingfor
forsome
some theorems
theorems associated
associatedwith
with the
thestrong
stronglaw,
law, Z.
Z. Wahrsch.
Wahrsch. Verw.
Verw.
Gebiete
Gebiete 19
19(1971),
(1971), 274-280.
274-280.
2.2. Y.
Y. S.
S. Chow,
Chow, Local
Local convergence
convergence of
ofmartingales
martingales and
and the
the laws
laws of
oflarge
large numbers,
numbers, Ann.
Ann. Math.
Math. Statist.
Statist. 36
36
(1965),
(1965), 800-807.
800-807.
690
T. P.
P. HILL
HILL
T.
3. L.
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A sharper
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sharperform
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the Borel-Cantelli
Borel-Cantellilenlma
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lawv,Ann.
Ann.
stroniglaw,
Math. Statist.
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(1965), 552-558.
552-558.
Math.
4. T.
T. P.
P. Hill,
Hill, Conditional
Conditionalgeneralizations
generalizationsof strong
which conclude
4.
strong laws
laws which
concludethe
the partial
partial sums
sums converge
convergealnlost
almost
Probab. 10
10 (1982),
Ann. Probab.
(1982), 828-830.
sure~v,
surelv, Ann.
I 'addition
5. P.
P. Levy,
Levy, Theorie
Theoriede
de I'
des variables
variablesaleatoires,
5.
addition des
alatoires, Gauthier-Villars,
1937.
Gauthier-Villars,Paris,
Paris, 1937.
P. Revesz,
Academic Press,
6. P.
The laws
laws of large
6.
Revesz, The
large numbers,
numbers,Academic
1968.
York, 1968.
Press, New York,
7. W.
W. F.
F. Stout,
Almost sure
sure convergence,
Stout, Alnlost
Academic Press,
7.
convergence,Academic
1974.
Press, New York,
York, 1974.
DEPARTMENT OF
OF MATHEMATICS,
DEPARTMENT
MATHEMATICS, UNIVERSITY
UNIVERSITY OF
OF LEIDEN,
LEIDEN, LEIDEN,
LEIDEN, THE NETHERLANDS
NETHERLANDS
Currentaddress:
address: School of Mathematics,
Institute of Technology, Atlanta,
Current
Mathematics, Georgia
Georgia Institute
Atlanta, Georgia
Georgia 30332
30332