Pub
.
Ma
.
UAB
Vol
.
29 Ns
1
Ab il
1985
FLUCTUATION
INEQUALITIESMITH
APPLICATIONS
TO
CONVERGENCE
AND
REGULARI'TY
OF
STOCHASTIC
PROCESSES
INDEXED
BY
[0,1]
q
.
An oniSin es
Blanc
Uni e si a
Au ónoma
de
Ba celona
Spain
Abs ac
.
Ex ending
esul s
o
Billingsley
and
Chen so ,
Bickel
&
Wichu a
p o ed
some
luc ua ion
inequali ies
o
p o-
cesseswi h
mul i-dimensional
ime
pa ame e
.
In
he
same
o de
o
ideaswe gi ehe e
an
ex ension
o
he
case
ha
he
ma ginals
o
he
con ol
measu e
a e
no
necessa ily
con inuous
.
Applica ions
o
his
esul s
o
ge
some
use ul
con e gen
ce
c i e ia
o
[0,1]
5
indexed
p ocesses
a e
gi en,
as well
as
a
heo em
on
egula i y
o igh s ochas ically
con inuous
p oce
sses
.
AMS
subjec classi ica ion
(1
.983)
.
60F05,
60005
Key
wo ds
and
ph ases
:
Fluc ua ion
inequali ies,
weakcon e gen
ce,
D[0,1]
a
- alued
andom a iables, egula i y
.
0
.
In oduc ion
.
In
(1)
P
.
Bickel
&
M
.
Wichu a
p o e
luc ua ion
inequali
ies
o
p ocesses
indexed
by
a q-dimensional
pa ame e
se ,
ex ending
esul s
o
Chen so
and,Billingsley,
(2),
(3)
.
He e
we ex end hei
heo em
3
o
he
casewhe e
he
ma ginals
o
m
a e
no
necessa ily
con inuous
.
Bickel
&
Wichu a
(op
.
ci
.
pg
.
1665,
inal)
announce
apossible
ex ension
o
he
case ha
m
dependson
n,
and he
measu esmn con e ge
weakly
o
a
measu e
wi h
con inuous
ma ginals
.
Ou
ex ension
has
a
di e en
cha ac e
:
m
will
be
ixed
(independen
o
n),
we
will
suppose
ins ead ha p ocesses
in
ques ion
ha e
independen
inc emen s,
and he
cons án s
ha appea
in
hei heo em
1
will
depend
on m,q,y
and
P,
in
ou
case
.
This
is
he
con-
en o poin
2
.
Poin
3
is
de o ed
o
gi e
applica ions
o
he
luc ua ion
inequali ies
o
he
con e gence
o
p ocesses
indexed
by
[0,1]
q
.
A poin
4
we
see
an
applica ion
o
he
egula i y
o
p ocesses
wi h
independen
inc emen s
o e
[0,1]
°
-
.
On
his
la e
esul
i
is~wo hy
o
say
ha
R
.
Mo -
k enas
(6),
using
Dynkin-Kinney's
ype
condi ions,
p o es
ha
all
p ocesses
wi h
independen inc emen s
ands ochas ically
con inuous
ha e
e sions
in

D
[0,1]
q
.

Ou Thm
.
(4
.1)
is
no
enclosed
in
his
esul
because
we onlyimpose igh
s ochas ic
con inui y
.
1
.
De ini ions
and
p e ious esul s
.
No a ion
is
much
as
in

(1)
.

Le

q

be
a
posi i ein ege
and

T1,TZ,
.
. . .
T
q
84
subse s
o
(0,1]
each
o
which
.con ains
0
and
1,
and
is
ini e
o

[0,11
.

Le

{X
} E
T

be
a
s ochas ic
p ocess
indexed
by
T
=
T
1
x T
Z
x
.
.
.x
T
q
,
wi h
alues
in
a
no med
space
(E,
I
.I)
.
We
suppose
X
is
sepa able
and
anish
on
he
lowe
bounda y
o
T,
ain T,
i
.e-
.
he
poin s
o
T
ha ing
some
coo dina e
equal
o
0
.
Fo
each

p,

1
<p
<q,

and
each

E
T_

we de ine
P
n
X P)
:
T
1
x
. .
.x
Tp x
.
.
.x
T
q
-
E

by
(P)
X
(
l
,. . . .

l
,

,
. .
.

)

=
X( l~
...
. p-l, , p+l
.
.
., q)
.
P'
P+1
q
I

s
6
9
u

in

T
,

we de ine
P
m

(s, , u)
(x)

=

min(11x
(P)
-

x
(
P
)
II,IIx
(
P
)
-

x(P)II)
p

su
Whe e
II
.
II
is
he
sup emum
no m
.
De ini ion
(1
.1)
:
M"
(X)

=
sup
{mp
(s, ,u)
(X)

:

s< <u
;

s,
,uET
P
}
M'
,
(X)

=

máx

Mp
(X)
1
<p
Qq
M
(X)

=

sup

{
1
X( )
1

:

E
T}

0
men a y
.
The
ollowing
p oposi ion
is
e yuse ul
and
qui e
ele-
q
P oposi ion
(1
.2)
:
I l
q =
(1,

l),
hen
q
M
(X)

<

E

M"
(X)

+

I
X(1

)I
p
=1P

q
5

q
.M"
(X)

+

I
X
(1
q
)
I

0
We
say
ha

B
C
T

is
a
block
i
q
B
=
II
(s
]
p=1
p p
we
also
w i e
B =
(s, ]
whe e
s=
(s
1
,..
.,S
)
and
4
Deno e
X(B)
he
ec angula
inc einen
o
X
o e
he
block
B, i .e
.
:
q
1

1

1

q-
El
p
X
(B)

=

E

E

.
. .

E

(-1)

P~

X(s
+e

( -s

),
. .
.,S
+
e

(
-s

))
el=0 e
2
=0

e
q
=0

1

1

1

1

q

q

q

q
TIe
say
ha
X
has
independen
inc emen s
i
X(B)
and
X(C)
a e
independen
andom
a iableswhene e B
and
C
a e
disjoin blocks
.
De ini ion
-
(1
.3)
:

We
w i e

X
EC
m
(P,y)

i

X

has
i
independen inc emen s
and
P
{
I
X
(B)

(m
(B)
)p,

a
x
>
0
o all
B
C
T,
block
o
T,
whe e
y
and
p
a e
ixed
posi i e
eals,
and
m
is
a
ini e
measu eo e
T
.
anishing
o e

a
in
T
E iden ly
i

XEC
m
(p
,y)

hen
he
pai

(X,m)

belongs
o
C(2p,2y)
in
he
sense
o
Bickel
&
Wichu a
(1)
.
Theo em
(1
.4)
:

I

(X,m)
E
C(P
.,'Y)
,
i.e
.

i
o
all
pai
o
disjoin
blocks
B,C
o
T
we ha e
hen

d
X
>
0
P{IX(B)I
>X,

IX(C)I

X}
:
Q
X
-y (m
(BUC))
p
,

yx>
0
o
all

p,

1
Sp
<q,

and
P{MB
(X)

}6K
q
(P
Y)

-y
(m(T)
)
R
P
{M"
(X)
>
T}6L
q
(p,y)

X
-y
(m(T))
p

O
This
is
heo em
1
o
Bickel
and
Wichu a(1)
.
In oduc ion
o
he
ollowingmoduli
is
sugges ed
by
he
iden i ica ion
D
q
=

D
(I
0,
l)
q
;

R)

=

D([0,11
;

D
q_1
)

n
0
De ini ion

(1
.5)
:

I

x
E
D
q
and

S
>
0

we
de ine
w"
(P)

(S)

=

sup

min
(II
x
(p)
-x
(p)
II
,II
x
(P)-x(P)
II
)
x

G
u
s,
,uETp
s
S -<u, u-s
5
S
w"
(S)
=
máx

w
.,
(P)
(S)
x

x
1-<P
! ~q
In
wha ollows
we
shall
also
need
he
ollowing
esul
on
igh ness
in
he
space
(D[O,l]
q
;
D
q
),
whosep oo
may
be
ound
in
Neuhaus
(7)
.
Theo em
(1
.6)
:
A
sequence
{Pn}n=1

o
p obabili y
measu es
on
(D[0,1
q,
Dq
)
is
igh
i
and
only
i
:
i)

Fo all

i?
>O,

he eexis s

aER

such
ha
Pn
{x
:
sup
1x( )I
>a}-< 1,

o
all
n>1
.
ii)
Fo all

e
>O,

77
>O,

he e
exis

S, 0
<
S<l,

and
such ha
o

n
>_n
.
Pn
{x
:
w
,
(S)
>e
}
S1
2
.
Fluc ua ionineguali ies
Theo em
(2
.1)
:

The e
exis s
a
cons an
K,
K =
K(q,R,y,

m(T)),

such
ha
o all
p ocess

XE
Cm(0
y),
(see
De
.
(1
.3)),
is
J
p
[
0,11

Q
m
P
[
0,1]1
o all

p, 1
<
p
-<q,

whe e

Jp[0,1]

is
he
maximum
jumpo
P{M"
(X)
>
ñ}<K(?~
47

V

X
27
)

(m
P
[
0,1] )2R

I1

-
p
he
dis ibu ion
unc ion
Fm
o
he
p- h
ma ginal,
m
p
,
o
P
m,
and
means
" he
g ea es
o "
.
P oo
:
S ep
1
.
g
=
1

and

T

ini e
.

Le

0 =
o
<
1
<
. .
.
<
m=
1
be
he
poin s
o
T
.
De ine
he
p ocess
m-1
Y
(U)

=

iE0

X
(
i
)
I[
i

i+1)
(u)

+

X
( m
)
I{
m}
(u)
o e
10,11
.
Then,
i
i-1
<8< i< h< < h+l< k<u< k+l
=
7~
-27
(
2
:

m{
j
}m{
j
,})
R
<
j
=i
.j
=
h
+1
h
<X
-27
[(
E
m{
.
} (
E

m{
.~}))
0
A
(
E

m{
.~}
(

m{
M)01
5
j=i

7

j=h+l

7

j=i

7

j
=1

7
<
x
-2'Y

[(m
(T)

-

J
m(T)
)
k
E
m{
.}l
R
j=i
7
k-1
C

27
(m
(T)
-
Jm
(T)
)~

(

E

2
m
{
j
} +
m{ k}-m{ i})R

=
j=i
--
a
-27
(m (T)
-Jm
(T)
)l)

(
k
E
1
2m{
j
}
+m{
k
}

-1
2
;
1
2m{
j }

-
m{ i}
)Q

=
_
X-27
(m
(T)

-

Jm
(T)
)0

(F (
k
)

-

F
(
i
)
)Q
<
<
X-27
(m (T)
-

J

(T)
)R
(F
(u)

-

F (s)
)Q
m
whe e
F,
con inuous,is de ined
by
he
ela ions
F(0)

=
0,

F(
j
)

-
F( j-1)

=
m{
j }

+
m{
j_1}

and
is
linea
o e
he
in e als

[
j
.
~-1
,

j
Hence,
he
p o es
Y,
oge he
wi h
he
measu e,
m',
associa ed
o
he
dis ibu ion
unc ion
F'
=
(m(T)
-
Jm(T))F,
belongs
o
C(R,27)
.
By
heo em
(1 .4)
we ha e
90
P{ML
(X)

}
=
P{M1
(Y)
T

}

KX
-2^i
(m' (T)
=

K
~
27

(m
(T)

-

J

(T)
)R

(F
(1)
)R
6
m
<
20

K
X
-27
(m
(T)
)R

(m
(T)

- J

(T)
1R
m
whe eweha eused

F(1)
<
2m
(T)
.

Thisp o es
he
heo em
in
his
case
.
S ep
2
.
g
=
1,

T =
[0,11,
m

a bi a
.

Le
0
=
o
<
1
<
. .
.
<
m =
1,

and

Y

he
p ocess

X

es ic ed o
{
, . .
.,
}
.
o
m
De ine
u
as in
s ep
4,
p oo o
heo em
1,
in
Bickel
&
Wichu a
(1)
:
u

{
J
.}

=

m(
J
.
-l
,
7
.]

i

j
ól,
u
{
o
}
=
0
.
Then

YE
Ci

as

a p ocess
o e

{
o
,
i . . .
m
} .
S ep
1
now
implies
J{
,
, . .
.,
}
R
P{ML
(Y)>X}<

X-2yK(u{ o
. l,
. . . .
m
})
2R

1
-

u

o

i

m
í

u
(
o
.. . . .
m}
K

X-27
(m
(T)
)
20

1

Ju[0
.1]

R
í
-
m(0,1]
I
now
we
ake
limi
when

m
--
"
hese
{ o, l,
..
., m}

inc easing
o
a
dense
subse
o
[0,1]

ha
con ains
he
poin so
discon inui y
o
F
,
we ob ain
(by se-
pa abili y)
:
(

]
P{M1
(Y)

]
X}

K
X-
2'y
(m(T)
)2R
11-

J
m
(
0, 1
m
(T)
S ep
3)
q
>-2,
T

and

m

a bi a y
.

We know ha
he
heo em
is
ue
o
q =
1
.
We
now
will
show
ou
esul
o
be
ue
o
p =
1
;
o
o he
p
he
a gumen
is
he
same
.
Like
in
s ep
5
o Bickel
&
Víichu a's
p oo
o heo-
em
1,
he
key
poin
is
ha
he
e sion
o

q=
1
o
ou
heo
em
wo ks
o he
unc ion
alued
p oces

{X
(l)}
E
T

To
1
Wi h
espec
o
ii)
:
a)
is
a
consequence
o
hm_
(3
.1),
c)

is
he
hypo hesis
iii)
.
Le
us
see
i)
.
98
Hence
:
P{
sup
ET

I
x
( )
I

>
a

}
=

P{
sup
II
x
(
P
)
II
>
a}
E[0
.11
P{sup
ET

I
(Xn) l
>
a
}
=
P
{

sup

II
(X
n
)
l
.)
II
>
a
}
E[
0,11
<P{w"
(1)
(S)
>
1
}
+P{máx

sup

I
(X

)
(1)

(
*
)
I
>
a

}
x

n

o
n

l
IQ i
<,
k

*
E
T
2x
.
. .
x
Tq
(1)

k
SP{w"

(8)
>1
}

+
E

P
{

sup

I

(x

I>
a
o
}
.
n

i=1

*
ET
2
x
. .
.
xT
q
Now
becauseo
P{w«X
n
)
(1)
[
1 - ó,l)
>
e,

o
some
p'
,

2
<p'
<q
}
6
.
1
6P{wXP
)
[1
-
8,1)>
e,
o
some
p,
16p<q
}
n
he
p ocesses
.
(Xn) l)
sa is y
i), ii)
and
iii)
o
ou
i
heo em,
i
=
1,2,
. . .
.k
.
By
induc ion
hypo hesis
he e
exis s
{
a
i
}
k
such
ha
P{sup
*I
(X,)
(1)
I
>
a
i }
6
n/2k
.
Gi en

n
>O,

le

S
>
0

be
such
ha
P{WX(1)

(S
)
>
1
}

X1/2
.
n

choose

0
=
0
<
I
<
. . .

k =
1

such ha

i -
i-1
<

ó

and
a
=
máx

a
.
o

i
.
.k
Then
i
a =
a
+ 1
0
P{sup
ET
I
(X
n
)
1
>a
}
<
n
Thisp o es
ha
i),
ii)
and
iii)
imply
i)
o
hm
.
(3
.2),
by
induc ion
on
q
.
0
E
<
S0
I
only
es s
o
e i y
condi ion
b)
o
ii)
.
By
induc ion
hypo hesis
(Xn
)S
P)
.W
;
(X)á P)
.
Hence,
II
(X
n
)
6
(
p
)
II
-~
II
(X)
s
(P)

also
.

Now
obse e
ha
as
a
consequence
o
he
igh
con inui y
o

(X) (P)
and
Gi en
posi i e

n

and

e,

le

S
0
>
0

be
such
ha
i
Then
II
(X)8(P)II
ó
0
P{II(X)(P)I1
:
e}<n/2
.
limsup

P{II
(xn
)
(
P
)
II
~e
}
<P{II
(X)
(P)
II
>e
}

n/2
.
n
-,
o0
{x
:
wXP
)
I
0,6)
>
4E
}
C
{X
:
w
"
(P)
(S
}
~
E
}
U
{x
:
II
x8(p)
-
X
0
(p)
II
~lE
}
ppose
:
y>0
.
Now
om
we
ge
:
lim

sup
__

P
n
{x
:
wXP)(
0,8
)

4e
}
:
!Z
17
Thisp o es
b)
and
he
heo em
.
In
applica ions
qui e
equen ly
we
don' know
ha
X
E
D
.

I
is
henuse ul

o
ha e
he
ollowing a ian
o
he
q
p e ious
hm
.
.,
whosep oo
equi es
he
same
a gumen
as
abo e
.
Theo em
(3
.4)
:
Le
00
be
as in
hm

,
(3
.3)
.
S
i)
The
ini e
dimensional
dis ibu ions
o
X
n
a e
weakly
con e gen
and
lim

lim
sup

P
{x
:
IIx
(
P
)
II
>E}
=0
Slo

n-'
oo
n
o all

E
>
0

andall

p,

1
<p<q
.
ii)

X
n
E
Ci
(Q
y)
,

n =
1,
2,
.
..,

o
some

Q
>112

and
iii)
lim

lim sup

P
n
{X
:W
(
P
)[
1-5,1)
>e,
o sume
p,
1<p<q}
=
0
640

n--~
o all
e
>O
.
Then

{P
n =
L(Xn)}n=1

is
.weakly
con e gen

O
4
.
Regula i y
o p
oce
sses
wi h
i
ndlQenden
in
c emen s
.
Theo em
(4
.1)
:

I

XEC
°
'
(R
,Y)
,

whe e

R
>
1/2,

y>0,
i
.
hen

X

has
a
e sion
wi h
sample
pa hs
in

D[
0,1]q
.
P oo
:

Le

8
0
<
1/2
.

Fo

E=-
[
0,1] q

de ine
:
6
0
( )

_

( 1
,...
,
i-1
, iI
[S
o
,
1-So]
( i
)

+S
o
I
[
0

So]
(
i)
+
o all

i,

1
<i
<q
.
(
1-8
0)
1
(1
-50,1]
(
i
),

i+l
,....
q
)
S
( )
=
((1
-250
)
1
( 1
0),
".
.,
(1-2
ó)
-1 (
i
-
S0
),
. .
.,
(1-25
0
)
1
(
q_80)
)
0
8
( )

_

(
8

0

S
0
0

a

0
. .
.a

s

)
( )
.
o

0

0

0
We
i s p o e
ha
he
p oces

Y =
X
S
( )
has
a
e sion
wi h
sample
pa hs
in

DIO,
llq
.

0
Obse e
ha

YE
Cm

i
1
m
(
.)
=
m(
8
(.
n[8o,1-8o]q))
0
on

10,
1]
q
,
Fo
each
n
we
de ine
a
p oces
Y
n
on [0,1,q, cons-
an o e each
ec angle
o
he
dyadic
ne
o
o de
n,
and
equal
o
he
alue
o
Y
a
"sou h-wes "
e ex,
i
.e
. :
o
all
E[
(i
1 -
1)2
-n
,i1 2
-n
)x
. .
.x[(i
q -
1)2-n,ig2-n),
1
.
<
i 1
6
2
n
, . .
.,

15

i
g

<2n
.
a gumen
like ha
in
he
p oo
o
hm
.
(3
.1)
shows ha
(4
.1 .1)

lim

limsup

P{w"

(ó
)
>
e
}
= 0
64
0

n
-

0o

Yn
lo

-
.
l l

1

I1
a
i

n-

c

~

.
In
ac
:
I Z

is
de ined
o e
T
*
=
[0,1]
P-l
x
[a, ]
x
n
x
[0,1]g
-
P

om
Y
n
,
as in hm_
(3
.1)
Y
n
is
de ined
om
X
,
Z
m(2)

ep esen s
he
es ic ion
o
Z

o
he
dyadic
ne ,
n
n

n
Tm(2)
,
o
T*,
and
(m)

is
de inedo e
Tm*(2)

like
he
o
s ep
2,
in
he
p oo
o
hm
.
(2
.1),
hen
:
Y
n
( )=Y((i
1
-
1)2
-n
,.
.
. .
(i

-
1)2n)
g
00
We
show
ha
{Yn}n=1
is
a
igh
sequence
.
Fi s ,
an
P{M"
(Z
)
>
X}
=
lim

P{M"
(Zm(2)
)
>ñ
}
S
P

n

m
-s
00

P

n
,
T
]
-IR
J
m
(a
'
lim

K
X-4y
(
m
(a~
1)
2
01
-

P

=
m
-~

00

P

m(a,T
]
j
]
-_
K
X_47
m
2Q
(a ]

1

-JP(a,
P

mp(a
.7]
whe e
Hence,
{Y
n
}
sa is ies
(3
.1
.3),
and
now
all
ollows
as
in
hm
.
(3
.1)'s
p oo
.
I

1-k
<b

and

T
k(2)

deno e
hese
o
poin s
o
he
.2-k
-dyadic
ne
in

T =
[
0,1]
q,

hen
sup ETIYn( )I
<
max ET

IYn( )I+qw'Y
(s)
k(2)

n
Mo eo e ,
obse e
ha
he
a iables
max
E

I
Y

( )
I
,

n
= k,

k
+

1,
...
Tk(2)
n
a e
iden icallydis ibu ed
.
This,
oge he
wi h
(4
.1 .1)
gi es
condi ion
i)
o
ou
hm
.
(3
.2)
.
Besides, {Yn}n=1sa is ies
b)
and
c)
o ii),
hm
.
(3
.2),
by
cons uc ion
.
Hence,
{Y
}°°

is
igh
.
I
W
is
he
weak
limi
o
n
n=1
some
subsequence, heni
is
easy
o
see
ha
W
is
a e sion
o
Y,
looking i s
a
dyadicpoin s,
and
app oaching
hen
.
any
poin
by dyadics
.
The
applica ion
g
^
being
bijec i e
and
con inuous
be ween
[S . ,l
-S
o
]
q
and
[O,l]q,
and
X
=
Y(Póo)-1( )
,
he
heo em
is
p o ed
.
0

Rema ks
and
commen s
.
a)
I
willbe e y
in e es ing
o
ge
a
esul
like
hm
.
(2
.1)
o
p ocesses
whoseinc emen s
a e
no
necessa ily
independen
.
I
don' IMow
a
p esen
how
o do
his
.
b)
All
p e ious
esul sex endeasily
o
[0,_)
q
-indexed
p ocesses
usina well
known
esul s
on
D[0,-)
q
(see

B
.G
.
I ano
(5))
.
c)
Using
abo e esul s
and
someo he s,
(which
cons i u-
e
my
Doc o al
Thesis,
as
p esen ed
a
he
Uni e si a
Au ónoma
de
Ba celona,
Spain),we
can
p o e
he
Cen al
Limi
Theo em
o
p ocesses
ha admi
a
ep esen a ion
as s ochas icin e-
g als
w
. .
.
.Lé y
p ocesses
wi h
mul idimensional
ime
pa ame-
e
.
Thiswillappea
elsewhe e
.
d)
Finally
I
wan o
exp ess
my indeb ness
and
g a i ude
o
P o esso
E
.
Giné,
ha
sugges ed
his
p oblems
o
me
and
has
gi en
e icien
help,
whene e
needed
.
REFERENCES
(1)
Bickel,
P
.J
.
&
Wichu a,
M
.J
.
(1971)
.
Con e gence
c i e ia
o
mul ipa ame e
s ochas ic
p ocesses
and
some
applica ions
.
The
Annals
o
Ma hema ical
S a is ics
ol
.
42
.
Ns
5,
1656-1670
.
(2)
Billingsley,
P
.
(1968),
"Con e gen e
o
P obabili y
Measu-
es"
.
John-Wiley
&
Sons
.
104
(3)
Chen so ,
N
.N
.
(1956)
.
Weak
con e gence
o
s ochas ic
p o-
cesses
whose
ajec o iesha eno
discon inui
ies
o
he
second
kind
and
he
"heu is ic"
,
app oach
o
he
Kolmogo o
-
Smi no
es "
.
Theo
.
P obabili y
Appl
.
1,
140-144
.
(4)
Giné,
E
.
&
Ma cus,
M
.B
.
(1983)
.
The
Cen al
Limi
Theo em
o
.
s ochas ic
in eg als
wi h
espec
o
Lé y
p ocesses
.
The
Annals
o P obabili y,
ol
.ll,
Ns
1,
58-77
.
(5)
I ano ,
B
.
(1980)
.
The
unc ion
space
D((o,oo)q
;E)
.
The
Canadian
Jou nal
o
S a is ics
.
Vol
.
8,
Ná
2,
179-191
.
(6)
Mo k enas,
R
.
(1980)
.
On
weak
compac nesso
he
se s
o
mul ipa ame e
s ochas ic
p ocesses
.
"S ochas
ic
Di e en ial
Sys ems"
.
Lec u e
No esin
Con ol
and
In o ma ion
Science,
25
.
(7)
Neuhaus,
G
.
(1971)
.
On
weak
con e gence
o s ochas ic
p o-
cesses
wi h
mul idimensional
ime
pa ame e
.
The
Annalso
Ma hema ical
S a is ics,
ol
.
42,
Ne
4
1285-1295
.
Rebu
el
11
de
duembne
del
1984
Uni e si a
Au ónoma
de
Ba celona
Facul a
de
Ciéncies
Depa amen
de
Ma emá iques
Bella e a
-
Ba celona
SPAIN