Pub
.
Ma
.
UAB
N° 22
No
.
1980
Ac es
VII
JMHL
SUMS
OF
INDEPENDENT
RANDOM
VARIABLES
AND
SUMS
OF
THEIR
SQUARES
E a is
Giné
Seccidde
Ma emá iques
Uni e si a
Au ónoma
de
Ba celona
Le
{Xni
:i=
1,
._k
n
,
n e
YO
be a
iangula
a ay
o
ow-wise
inde-
penden
andom
a iables,
Sn=
EJXnj
he
ow
sums
and
Tn=
E
j
x2j
he
ow
sums
o
squa es
.
Raiko
(1938)
p o ed
ha
S
con e gesweakly
o a
Gaussian
law
i
and
only i
Tn
con e ges
in
p obabili y
o
a
cons an
.
Hall
(1978)
shows
ha
i Sn
con e ges
o
a
Poisson
law
wi h
pa ame e
a,
hen so
does
T
n
.
In
his no e we
gi e
he
exac ela ion
be ween
igh ness
and
con e gen e
o
{L(Sn)},{L(Tn
)}
and
{min(1,x
2
)E
j
dL(X
ni )}
o
in ini esimal

a ays
;
he-
se
esul s
con ain
hose
o Raiko
and
Hall
as
pa icula cases
.
The
igh -
ness
ela ions
p o ed o
be
use ul
in
some
wo k
wi h
M
.P
.Ma cus
on
he
cen-
al
limi
heo em
in
C(S)
.
I
acknowledge
P o
.
M
.Ma cus
o he
co espon-
dence
ha
led o
his no e
(as
a
byp oduc )
.
The
no a ion
will

be as
ollows
:

{X
nj
:

j
=1_
.,k
n
,

n e
i}
will

be
a
iangula
a ay
o
ow-wiseindependen
andom
a iables
({X
nj
}
o
sho ),
Sn
=E
.X
n
,
T
=E
.X
2
,
X

=
X
I

<
T}'
Sn,T-EjxnjT'
(T>0),
J

j

n

J

nj

ni
,
ni

{IX
n
i j_
and
{Xnj
}
will
deno e
independen
symme iza ions
o
{X
nj
}
.{X nj
)
is
in ini
-
esimal
i
lim
n
max
j
P{IX
ni
1
>
E}
=0
o
all
E>0
.
IJe
e e
o
Gnedenko
and
Kolmogo o
(1968,
Theo em
25
.1)
o
o
A aujo
and
Giné (1980,
Theo em
2
.4
.7)
o he
gene al
cen al
limi
heo em
on
he
line(CLT)
.
The ollowina
is
ou
main
obse a ion
.
I
elabo a es
on
exe cise
2
.5
.3
o
A aujo
and
Giné (1980)
and
i s
p oo
is
inspi edon
hei p oo
o
he
con e se
CLT
.
heo em
1
.
Le
{Xnj
}
be a
iangula
a ay
o ow-wise
independen
's
.
Then
{L(E
i
xn
i
)} is
igh
i
and
only
i
he
amily
measu es
is
uni o mly
bounded
and
igh
.
d
n
(x)=
min(l,x
2
)E
j
dL(X
nj )(x)
12
7
P oo
.
Assume
{L(E~X2~)} igh
.
Then by
posi i i y,
so
a e
{L(EiX2iT)}n-1
and {L(Xni
)}n,J
.
The
con e se
Kolmogo o
and
Lé y
inequali ies
gi e
whe e
c
T
d
is
a
ini e
cons an
o
each
T,
d
> 0
.
The igh ness
o
{L(Ei(X2jT)-)} he e o e
implies
sup
n
E(E
j
(X2~
T -
c
_

<
hence
ness
o
{L(E
.X
2
-E
.EX
2
)}
by
Chebyshe
.
{L(E
.X
2
)}
being
igh , we con-
,1
niT
J
ni
T
~
niT
elude
(1)

sup
n
E~EX2
jT
o
al]
T
>O
.
As
is
well
independen symme ic,
hen
o
al] d>
0(as
we
can
conclude
Since
which
E(E
i
(X2
jT-EX2iT
))
2
I
we
he e
exis s
T
>
0
such ha
This
p o es ha

{E
i
L(Xnj)lIxlI>
T1/2}
ce by
(1
,
),

so is

{min(1,x
2
/T)E
i
L(X
ni
)}
some
T>
0
hese
measu es
a e
uni o mly
o
each
T
>O,
in
pa icula
o
- =
1
.
Con e sely,
assumenow
ha
(
n
}
o
al
l

T,

>
0,
12
8
<
CT,d/(1-2P{IEi(X2iT)-j'>d}1
known,
Lé y's
ineqiali y
gi es
ha
En
=1
P{In
i
I
> d} <

-log(1-2P{IE
i
n
i
l
>
6»
obse ed
by
Felle
(1971),
page
149)
.
Hence,
ha
he e
exis
B>
0
and
xni
e
IR

such ha
supnEjP{IX2J-xni1
>S}
<
1/2
.
{L(X
2

is
igh ,
he e
exis s
M>
0
such ha
sup_
;P{X?
;>
M}<
1/2,
1J
.
1,J

_

11,J
11J
implies
ha
Ix
ni
¡
<
M+S
.
So,
he e
exis s
T
>
0
such ha
sup
n E
i
P{X2
i>
T}
<
1/2
.
i
{n
i
}
a e
using
Fubini
apply
his o
independen
copies o
Sn
,
e1N,
we
conclude
ha
sup
n E
j
P{X2
i
>T
} <
1/2
.
P{T
n
> }<

P{E
i
X2
iT
> ,

IXn
il
<
T,
J=1,
. .
.,kn
}
he
igh -
is
uni o mly
bounded
and
igh ,
hen-
I is
i ial
o
see
ha
i
o
bounded
and
igh ,
he
same
is
ue
is
uni o mly
bounded
and
igh
.
Then
+EiP{
I
XnJ
l
>
T}
<

EjEX2iT/ +EiP{
I
Xn
.l
>
T}

.
Gi en
e
> 0
choose
T
>
0
such ha
he
las
sum is
no
g ea e
han
e/2
and
hen
such
ha
E
j
EX
2
/
<e/2
.
Hence,
{L(T
n
)} is
igh
.0
Since
(2)

E(Xni-EXnj1)
2
1{IXnJI
<l}-EX
2
njl
=
he
p e ious
heo em
oge he
wi h
heo em
2
.45
in
A aujo
and
Giné(1980~gi e
:
Co olla y
2
.
Le
{Xnj
}
be an
in ini esimal
a ay
such
ha
(3)

sup
n
Ej (EX
ni1
)
2<
m
.
=-(1+p{IXn
.i1
>
l})(EXnJl)2,
Then,
{L(Sn-ESn,T)}
is
igh
i
and
only
i
{L(E
j
X2
j
)}
is
.
Rema k
.
Condi ion
(3) is
sa is ied
i
:
(a)
{X
nj
}
is
symme ic,
bu
in
his
case
i
is
no
necessa y
o
assume
in-
ini esimali y
(use
A aujo
and
Giné (1980)
.
Co
.
2
.5
.7),
and
(b)

o
{Z
ni=
Xn
j-EX
njT
},
any
T
> 0,

i
ei he
{L(S
n
-ESn,T
)}
o
{L(E
i
(X
ni
-EX
njT
)
)}
a e
igh
o
some

T>0
.
Le
us
see
i
o
T
=
1
1
EZnj1l <
l1'IXni1
<1ZnidPi
+(1+
[EXn
j1l)P{IX
n
jl
>
1-IEXnjll}
=
¡EXn
jllp{IX
n
jl>
1}
+(1
+¡EX
n
jll)P{IX
n
jl
>
1-IEXnjll},
and
since
max
j
IEX
njl
l->0
as
n-
by
in ini esimali y,
we
ob ain
ha
sup
n
E
j
(EZnil
) 2
< c
sup
n
E
J
P{IX
ni
1
>
1/2}
and
his
quan i y
is
ini e
i
ei he
one
o
he
wo
amilies
o sums
a e igh ,
by
he
p e ious
heo ems
.
So we
ha e
:

-
Co olla y
3
.

Le
{X
ni*
}
be
in ini esimal
.

Then

{L(S
n
-ES
n,d
)}

is
igh
i
and
only
i
{L(E
i
(X
ni
-EXnjd
)
2
)} is
igh
o
some
(all)
d
> 0
.
Nex we
examine
con e gen e
ela ions
.
We
will
le
T(x)
= x2
.
No e
ha
i
p
and
a e
a- ini e
Bo el
measu es
he
equa ion
oT
=P,
unknown,
has
a
uniquesymme ic
solu ion
and
a
unique
solu ion
suppo ed
by
R
+
(R
-
)
.
This
solu ion
will
be
deno ed
=
poT
and
i
will

be
symme ic
o
suppo éd
by
R
+
depending
on
he
con ex
.
Theo em
4
.
Le
{Xnj
}
be
an
in ini esimal
a ay
.
(a)
Assume
{Xnj
}
sa is iescondi ion
(3)
and
(4)

L(S
n
-ESn,a
)-
w
N(O,a
2
)*c
6
Poisp
12
9
o
some
o2> 0,
Lé y
measu e
)
.+
and
6
>0
such ha
p{-6,0=
0
.
Then,
(5)

L(E
i
X2
i
-E
i
EXñ
ja
)->
wc6
2
Pois(poT
-1
)
In
pa icula ,
i
condi ion
(3)
is
eplaced
b
.
y
he
s onge
condi ion
(6)

l
imn
Ej
EX2
ja
=
a
<
~,
hen
(7)

L(EjX2,j)->w6a*cal
Pois(poT-1)
(b)
Con e sely
i
he
X
ni
a e non-nega i e
(symme ic)
and
(8)

lim
ayOlim
nE
i
(EXnja
)
2 =
0,
hen,
he
ac ha
(9)

L(E
j
X2
)_w6a*ca2
Poisp
o
some Lé y
measu e
p
and
a
such ha p{-6
2
,6
2
}=
0,
implies
(
10
)
.

L(S
n
-ES
n,a
)-
w
N(O,o
2
)*c
6
Pois(poT),
9
whe e
o~
=a-
n

xdp(x)
.

(poT
is
symme ic
i
he
Xn
a e
symme ic
and
wi h
,,
suppo
in
h
+
i
he
X
n
,
a e
non-nega i e)
.
Also,
E~EX2~
a
->a
.
(b')

I
in

(9)

ji=
0,

hen

(b)

is
ue
wi hou
he
a iables
X
ni

being
non-
nega i e
o
symme ic
.
P oo
.

(a)

By
he
CLT

,

Ei
L(X
nj
)
I{Ixi
>8}
->wpI{Ix1
>61

i
w{-M}=
0
.
Then
i p{-6
1 /
2
,6
112
}=
0,
E
.L(X
2
)1

poT
-1
~

On
he
o he
J

ni
{~x~
>6}-
w

{ix1
>0*
hand,
because
by
condi ion
(3)
and
he
CLT

,
sup
nz
j
EX2
<
-
(see
(2))
.
Hence
(5)
ollows
om
he CLT
.

nil
(b)
Assume
now
ha
(8)
and
(9)
hold
.
Then
(3)
holds
°
and he e o eCo olla y
3
gi es
ha {L(S
n
-ESn,a
)}
is
igh
.
Bu ob iously
al]
he
subsequen ial
limi s ha e
he
same
Lé y
measu e
poT,

hence
E
i
L(X
nj)j
{
,
x
,
>6},
wpoTj{jxj>6}
13
0
lim64-0limnEj
E[XniI{X2J
<6}
-(EX
ni
I
{X2
i
<6})2l
1¡m
ay
01¡m
n
E~(¿
-
E(Xn
J
)
6
)E(Xn
J
)
6
=0
i
poT{-8,8}=
0
.
Now,
he
CLT

and
(9)
gi e
lim
nz j
EX2
=a,
and
he e o e,
condi ion
(8)
implies
limTyO
{lim}nEj(EXnjT)2)=
limTy,0
{lim}nEjEXniT
2
=
1
im
T+0,p{T
2 } =
0l
im
n
Ei
EXn
jT
=
1
im
TyO,u{-
2
}
=
0
[a-
1?
xdp(x)]
l
2
=a- 0
xdu(x)
. .
So, (10)
ollows
by
he CLT
.

(b')
also
ollows
om
he
CLT,
Gaussian
con e gence,
and
om
(ii)
wi h
p
=0
.
Rema ks
41)
Condi ion
(8) is
sa is ied
in
he
symme ic
case
and
also
o
Znj
= X
nj
-EX
nj1 in
gene al
( om
ema k
(b)
a e
Co olla y
3
we
ob ain
ha
i
{L(E
j
Xn
j
)}
o
{L(E
j
7ñ
j
)}
a e
shi
igh ,
hen
limn
E
j
EIZnjd
l
2 <
<

lim
nz j
(P{1
Xnj
1
>
6})`
<
lim
n
maxj P{lX
nj1
>8}
,
Ej(P{1X
n
j1
>
8}=
0)
.
(2)
Le
us
inally
ema k
ha
i
bo h {L(S
n
)}
and
{L(E
i
Xni
)}
con e ge,
hen
he
p- h
momen
o
IS
n
1
con e ges
i
and
only
i
he
(p/2)- h
momen
o
E
i
Xn
i
does
:bo h
condi ions
a e
equi alen
o
1im ~sapnziE1XniiPI{IXnj
1
> }=0
(de
Acos a
and
Giné
(1978))
.
Wi h his
ema k,
Theo em
5
con ains
he
esul
in
Hall
(1978)
as a
pa icula
case ( he
cases
p =
0
and
p =
x81)
.
(3)
I
is
also
clea
om
he
o egoing
ha
he
powe
2 is
basic
only
i
lim
nEj EX2
j
0
and
he
Lé y
measu e
u
(o poT
-1
)
gi es
posi i e
mass o
in e als
a bi a ily
nea
o
ze o
.
0 he wise
he
p e ious
esul s
hold
o
EjlX
n
j1

,

o any
p
>
0
(as
obse ed
by
Hall

(1978)

in
he
pa icula
case
p=x8
1 )
.
REFERENCES
1
.-
de
Acos a
,
A
.
and
Giné,
E
.
(1979)
.
Con e gence
o
momen s
and
ela ed
unc ionals
in
he
gene al
cen al
limi
heo em
in
Banach spaces
.
Z
.
Wah scheinlichkei s heo ie
e w
.
Gebie e
48,
211-231
.
2
.-
A aujo,
A
.
and
Giné,
E
.
(1980)
.
The
cen al limi
heo em
.
Wiley,New
Yo k
.

3
.-
Felle ,
W
.
(1979)
.
An
in oduc ion
o
P obabili y
Theo yand
i s
appli
-
ca ions
.
Wiley,
New
Yo k
.
4
.-
Gnedenko,
B
.V
.
and
Kolmogo o ,
A
.N
.
(1954)
.
Limi
dis ibu ions
o
sums o
independe
n
andom
a iables
.
Addison-Idesley,
Reading,
Mass
.
5
.-
Hall,
P
.
(1978)
.
On
he
duali ybe ween
he
beha iou
o sums o
inde-
penden
andom a iables
and
he
sums
o
hei
squa es
.
Ma h
.
P oc
.
Camb idge
.
Phil
.
Soc
.
84,
117-121
.
6
.-
Raiko ,
D .A
.
(1938)
.
On a
connec ion
be ween
he
cen al
limi
heo em
in
he
heo y
o
p oba
bili yand
he
la<
o
la cpnumbe s
.
Iz es ya
Akad
.
Nank
.
SSSR
(Se
.
Ma
.)