Limit theorems for radial random walks on Euclidean spaces with ...

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Oct 8, 2013 (3 years and 8 months ago)

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Limit theorems for radial random walks on Euclidean
spaces with growing dimensions
Waldemar Grundmann
TU Dortmund,Germany
Joint work with Michael Voit
Munchen,September 2012
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 1/16
Classical limit theorems:
xed state space Z
(S
n
)
n0
Markov chain on Z
Limit theorem for (functionals of) S
n
,n!1
Random matrix theory:
sequence of spaces Z
p
and measures 
p
2 M
1
(Z
p
)
mappings'
p
:Z
p
!Z
Limit theorems for'
p
(
p
) 2 M
1
(Z),p!1
Mixed Problems
sequence of spaces (Z
p
)
p
;mappings'
p
:Z
p
!Z
On each Z
p
a time homogeneous Markov chain (S
p
n
)
n0
Limit theorems for'
p
(S
p
n
) on Z for n;p!1in a coupled way
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 2/16
Classical limit theorems:
xed state space Z
(S
n
)
n0
Markov chain on Z
Limit theorem for (functionals of) S
n
,n!1
Random matrix theory:
sequence of spaces Z
p
and measures 
p
2 M
1
(Z
p
)
mappings'
p
:Z
p
!Z
Limit theorems for'
p
(
p
) 2 M
1
(Z),p!1
Mixed Problems
sequence of spaces (Z
p
)
p
;mappings'
p
:Z
p
!Z
On each Z
p
a time homogeneous Markov chain (S
p
n
)
n0
Limit theorems for'
p
(S
p
n
) on Z for n;p!1in a coupled way
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 2/16
Classical limit theorems:
xed state space Z
(S
n
)
n0
Markov chain on Z
Limit theorem for (functionals of) S
n
,n!1
Random matrix theory:
sequence of spaces Z
p
and measures 
p
2 M
1
(Z
p
)
mappings'
p
:Z
p
!Z
Limit theorems for'
p
(
p
) 2 M
1
(Z),p!1
Mixed Problems
sequence of spaces (Z
p
)
p
;mappings'
p
:Z
p
!Z
On each Z
p
a time homogeneous Markov chain (S
p
n
)
n0
Limit theorems for'
p
(S
p
n
) on Z for n;p!1in a coupled way
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 2/16
Example:radial random walks on R
p
Let Z
p
:= R
p
,Z:= [0;1[,'
p
:R
p
![0;1[,x 7!kxk
2
.
Fix  2 M
1
([0;1));Then for each dimension p there is a unique
radial 
p
2 M
1
(R
p
) with radial part ,i.e.,

p
is SO(R
p
) invariant and'
p
(
p
) = :
Consider i.i.d.random variables (X
p
k
)
k1
on R
p
with law 
p
Then (S
p
n
:=
P
n
1
X
p
k
)
n0
is called a radial random walk on R
p
with
law 
p
Problem:Limit theorems for'
p
(S
p
n
) = kS
p
n
k
2
for n;p!1?
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 3/16
Example:radial random walks on R
p
Let Z
p
:= R
p
,Z:= [0;1[,'
p
:R
p
![0;1[,x 7!kxk
2
.
Fix  2 M
1
([0;1));Then for each dimension p there is a unique
radial 
p
2 M
1
(R
p
) with radial part ,i.e.,

p
is SO(R
p
) invariant and'
p
(
p
) = :
Consider i.i.d.random variables (X
p
k
)
k1
on R
p
with law 
p
Then (S
p
n
:=
P
n
1
X
p
k
)
n0
is called a radial random walk on R
p
with
law 
p
Problem:Limit theorems for'
p
(S
p
n
) = kS
p
n
k
2
for n;p!1?
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 3/16
Example:radial random walks on R
p
Let Z
p
:= R
p
,Z:= [0;1[,'
p
:R
p
![0;1[,x 7!kxk
2
.
Fix  2 M
1
([0;1));Then for each dimension p there is a unique
radial 
p
2 M
1
(R
p
) with radial part ,i.e.,

p
is SO(R
p
) invariant and'
p
(
p
) = :
Consider i.i.d.random variables (X
p
k
)
k1
on R
p
with law 
p
Then (S
p
n
:=
P
n
1
X
p
k
)
n0
is called a radial random walk on R
p
with
law 
p
Problem:Limit theorems for'
p
(S
p
n
) = kS
p
n
k
2
for n;p!1?
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 3/16
Expected value and Covariance of 
p
:
Let X  
p
then:
E(X) = 0 2 R
p
and Cov(X) =
m
2
()
p
 I
p
2 R
pp
;
where m
k
():=
R
1
0
x
k
d(x),k 2 N.
Extreme case:p 2 N xed,n!1
Classical CLT on R
p
and projection'
2
p
:x 7!kxk
2
2
yield

n;p
:='
2
p
(
q
p
m
2
()n
 S
p
n
)
d
!
2
p
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 4/16
Expected value and Covariance of 
p
:
Let X  
p
then:
E(X) = 0 2 R
p
and Cov(X) =
m
2
()
p
 I
p
2 R
pp
;
where m
k
():=
R
1
0
x
k
d(x),k 2 N.
Extreme case:p 2 N xed,n!1
Classical CLT on R
p
and projection'
2
p
:x 7!kxk
2
2
yield

n;p
:='
2
p
(
q
p
m
2
()n
 S
p
n
)
d
!
2
p
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 4/16
General setting
(p
n
)
n1
a sequence of growing dimensions (i.e.p
n
!1)
 2 M
1
([0;1)) with nite second moment m
2
:= m
2
()

p
n
2 M
1
rad
(R
p
n
) corresponding radial probability measure on R
p
n
X
1
;:::;X
n
i.i.d.random variables on R
p
n
with law 
p
n
S
p
n
n
:=
P
n
1
X
k
radial sum on R
p
n
Consider the process


n
():= kS
p
n
n
k
2
2
n  m
2

n1
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 5/16
Theorem
Assume that  admits a nite fourth moment m
4
< 1.
If
n=p
n
!1;then
p
p
n
n

n
()
d
!N

0;2m
2
2

CLT I
If
n=p
n
!0;then
1
p
n

n
()
d
!N

0;m
4
m
2
2

CLT II
If
n=p
n
!c > 0;then
1
p
n

n
()
d
!N

0;m
4
m
2
2
+2cm
2
2

CLT III
M.Voit:CLT I holds for
n
p
3
n
!1;CLT II holds for
n
2
p
n
!0.
Proof:The proof is divided into two main steps:
STEP 1: with compact support;with method of moments
STEP 2:m
4
< 1;with truncation method
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 6/16
CLT by convergence of moments
Theorem (method of moments)
;
n
2 M
1
(R),(n 2 N) and  is determined by m
k
().Then
m
k
(
n
)!m
k
() =)
n
!
Consider the moments M
n
k
:= E(
n
()
k
).Then
M
n
0
= 1;M
n
1
= 0;
M
n
2
= n(m
4
m
2
2
) +2n(n +1)m
2
2
=p
n
M
n
3
=:::
Formulas become too complicated to detect a general limit pattern!
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 7/16
Sketch of proof:
Decomposition:Write

n
() =
n
X
i =1

kX
i
k
2
2
m
2

+
p
n
X
j=1
X
i
1
6=i
2
2f1;:::;ng
X
(j)
i
1
X
(j)
i
2
=:A
n
+B
n
Theorem (Convergence of A
n
and B
n
)
Assume that  has a compact support.
c:= lim
n!1
n
p
n
scale
lim
n!1
~
A
n
lim
n!1
~
B
n
CLT I c = 1 n > p
n
p
p
n
n

0
N(0;2m
2
2
)
CLT II c = 0 n < p
n
1
p
n
N(0;m
4
m
2
2
)

0
CLT III 0 < c < 1 n  p
n
1
p
n
N(0;m
4
m
2
2
)
N(0;2cm
2
2
)
Proof is based on
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 8/16
Sketch of proof:
Decomposition:Write

n
() =
n
X
i =1

kX
i
k
2
2
m
2

+
p
n
X
j=1
X
i
1
6=i
2
2f1;:::;ng
X
(j)
i
1
X
(j)
i
2
=:A
n
+B
n
Theorem (Convergence of A
n
and B
n
)
Assume that  has a compact support.
c:= lim
n!1
n
p
n
scale
lim
n!1
~
A
n
lim
n!1
~
B
n
CLT I c = 1 n > p
n
p
p
n
n

0
N(0;2m
2
2
)
CLT II c = 0 n < p
n
1
p
n
N(0;m
4
m
2
2
)

0
CLT III 0 < c < 1 n  p
n
1
p
n
N(0;m
4
m
2
2
)
N(0;2cm
2
2
)
Proof is based on
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 8/16
Lemma (moments of 
p
2 M
1
rad
(R
p
))
Let a = (a
1
;:::;a
p
) 2 N
p
0
and k:=
P
a
i
.Then
m
a
(
p
) =
Z
R
p
x
a
1
1
   x
a
p
p
d
p
(x) =
8
>
<
>
:
C
(
p+k
2
)
(
p
2
)
m
k
() if a = (2l
1
;:::;2l
p
)
0 otherwise
with the constant C:= C(k):= 2
k
a!=(a=2)! k!.
Proof:Since 
p
is radial,we have

p
=
Z
1
0
U
(r)
p
d(r) 2 M
1
rad
(R
p
)
where U
(r)
p
is the uniform distribution on the sphere S
(p1)
r
 R
p
with
radius r > 0.Therefore
m
a
(
p
) =
Z
1
0
Z
R
p
x
a
1
1
::: x
a
p
p
dU
(r)
p
(x)d(r) =
Z
1
0
m
a

U
(r)
p

d(r)
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 9/16
Lemma (moments of 
p
2 M
1
rad
(R
p
))
Let a = (a
1
;:::;a
p
) 2 N
p
0
and k:=
P
a
i
.Then
m
a
(
p
) =
Z
R
p
x
a
1
1
   x
a
p
p
d
p
(x) =
8
>
<
>
:
C
(
p+k
2
)
(
p
2
)
m
k
() if a = (2l
1
;:::;2l
p
)
0 otherwise
with the constant C:= C(k):= 2
k
a!=(a=2)! k!.
Proof:Since 
p
is radial,we have

p
=
Z
1
0
U
(r)
p
d(r) 2 M
1
rad
(R
p
)
where U
(r)
p
is the uniform distribution on the sphere S
(p1)
r
 R
p
with
radius r > 0.Therefore
m
a
(
p
) =
Z
1
0
Z
R
p
x
a
1
1
::: x
a
p
p
dU
(r)
p
(x)d(r) =
Z
1
0
m
a

U
(r)
p

d(r)
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 9/16
The Fourier transform of U
(r)
p
corresponds to the modied Bessel function


of the rst kind,namely
^
U
(r)
p
(z) = 
p
2
1
(r  kzk
2
);z 2 R
p
;where


(w) =
1
X
j=0
(1)
j
( +1)(w=2)
2j
j!(j + +1)
;jwj < 1;jarg(w)j < :
From this,using the relation
D
a
^
U
(r)
p
(z)



z=0
= i
jaj
m
a
(U
(r)
p
)
we conclude
m
a

U
(r)
p

=
8
>
<
>
:
r
2l
4
l


p
2

l
p
Q
j=1
(2l
j
)!
l
j
!
if a = (2l
1
;:::;2l
p
)
0 otherwise
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 10/16
Proof of CLT I:n=p
n
!1,supp() compact
Let k 2 N.The k-th moment
~
M
n
k
of
p
p
n
n
 
n
() is given by
~
M
n
k
=

p
n
n
2

k=2
k
X
l =0

k
l

E

A
l
n
B
kl
n

By the convergence of
p
p
n
n
 A
n
and
p
p
n
n
 B
n
we have
if l = 0,then

p
n
n
2

k=2
E

B
k
n

!m
k
(N(0;2m
2
2
))
if l 2 f1;:::;kg,then

p
n
n
2

k=2 

E

A
l
n
B
kl
n





p
l
n
n
2l
E

A
2l
n

|
{z
}
!0
p
kl
n
n
2(kl )
E

B
2(kl )
n

|
{z
}
!m
2(kl )
(N(0;2m
2
2
))
1
2
!0
By method of moments CLT I follows.
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 11/16
proof of CLT III:n=p
n
!c 2]0;1[,supp() compact
A
n
and B
n
are asymptotically independent,i.e.for 0  l  k:

1
p
n

k

E

A
l
n

E

B
kl
n

E

A
l
n
 B
kl
n

n!1
!0 (AI)
the k-th scaled moment
~
M
n
k
:=
1
n
k=2
E((A
n
+B
n
)
k
) converges to 0 if
k is odd and for k = 2j we have:
lim
n!1
~
M
n
k
= lim
n!1
j
X
l =0

2j
2l

E

1
n
l
A
2l
n

 E

1
n
jl
B
2(jl )
n

=
j
X
l =0

2j
2l

m
2l
(N(0;m
4
m
2
2
))  m
2(jl )
(N(0;2cm
2
2
))
For j 2 N,
2
1
;
2
2
2 R
+
we see at once that
j
X
l =0

2j
2l

m
2l

N(0;
2
1
)

m
2(jl )

N(0;
2
2
)

= m
2j

N(0;
2
1
+
2
2
)

:
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 12/16
Theorem (strong law of large numbers)
Assume that  admits a nite eighth moment m
8
< 1.
If
n=p
n
!1;then
p
n
n
2

n
()
n!1
!0
a.s.
If
n=p
n
!c  0;then
1
n

n
()
n!1
!0
a.s.
Proof:Let a
n
= n if n=p
n
!c and a
n
= n
2
=p
n
if n=p
n
!1.We calculate
Z
A
n
()
4
dP  C()n
2
and
Z
B
n
()
4
dP  C()
n
4
p
2
n
:
Hence,by Markov inequality,
0  P(j
n
()j    a
n
)  P

jA
n
j 
  a
n
2

+P

jB
n
j 
  a
n
2

 C()
4
a
4
n

n
2
+n
4
=p
2
n

= O(n
2
)
Now,by Borel-Cantelli lemma,P(j
n
()j    a
n
i.o.) = 0 for each
positive .
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 13/16
Theorem (strong law of large numbers)
Assume that  admits a nite eighth moment m
8
< 1.
If
n=p
n
!1;then
p
n
n
2

n
()
n!1
!0
a.s.
If
n=p
n
!c  0;then
1
n

n
()
n!1
!0
a.s.
Proof:Let a
n
= n if n=p
n
!c and a
n
= n
2
=p
n
if n=p
n
!1.We calculate
Z
A
n
()
4
dP  C()n
2
and
Z
B
n
()
4
dP  C()
n
4
p
2
n
:
Hence,by Markov inequality,
0  P(j
n
()j    a
n
)  P

jA
n
j 
  a
n
2

+P

jB
n
j 
  a
n
2

 C()
4
a
4
n

n
2
+n
4
=p
2
n

= O(n
2
)
Now,by Borel-Cantelli lemma,P(j
n
()j    a
n
i.o.) = 0 for each
positive .
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 13/16
Euclidean case,higher rank:Z
p
= M
p;q
(F)
F = R;C;H with real dimension d = 1;2;4
M
p;q
:= M
p;q
(F) with scalar product hx;yi:= <(tr(x

y))
U
p
:= U
p
(F) acts on M
p;q
by (u;x) 7!ux
U
p
x = U
p
y () x

x = y

y,hence M
U
p
p;q
'
q
'
p
:M
p;q
!
q
;'
p
(x):=
p
x

x:
Example:F = R,q = 1:M
p;1
'R
p
,'
p
(x) = kxk
2
,
1
= [0;1[.
Similar to q = 1:
Fix  2 M
1
(
q
);Then for each dimension p there is a unique radial

p
2 M
1
(M
p;q
) with radial part  2 M
1
(
q
),i.e.,

p
is U
p
invariant and'
p
(
p
) = :
(S
p
n
)
n
radial RW on M
p;q
with radial distribution  2 M
1
(
q
)
Problem:Limit theorems for'
p
(S
p
n
) for n;p!1
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 14/16
Euclidean case,higher rank:Z
p
= M
p;q
(F)
F = R;C;H with real dimension d = 1;2;4
M
p;q
:= M
p;q
(F) with scalar product hx;yi:= <(tr(x

y))
U
p
:= U
p
(F) acts on M
p;q
by (u;x) 7!ux
U
p
x = U
p
y () x

x = y

y,hence M
U
p
p;q
'
q
'
p
:M
p;q
!
q
;'
p
(x):=
p
x

x:
Example:F = R,q = 1:M
p;1
'R
p
,'
p
(x) = kxk
2
,
1
= [0;1[.
Similar to q = 1:
Fix  2 M
1
(
q
);Then for each dimension p there is a unique radial

p
2 M
1
(M
p;q
) with radial part  2 M
1
(
q
),i.e.,

p
is U
p
invariant and'
p
(
p
) = :
(S
p
n
)
n
radial RW on M
p;q
with radial distribution  2 M
1
(
q
)
Problem:Limit theorems for'
p
(S
p
n
) for n;p!1
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 14/16
Euclidean case,higher rank:Z
p
= M
p;q
(F)
F = R;C;H with real dimension d = 1;2;4
M
p;q
:= M
p;q
(F) with scalar product hx;yi:= <(tr(x

y))
U
p
:= U
p
(F) acts on M
p;q
by (u;x) 7!ux
U
p
x = U
p
y () x

x = y

y,hence M
U
p
p;q
'
q
'
p
:M
p;q
!
q
;'
p
(x):=
p
x

x:
Example:F = R,q = 1:M
p;1
'R
p
,'
p
(x) = kxk
2
,
1
= [0;1[.
Similar to q = 1:
Fix  2 M
1
(
q
);Then for each dimension p there is a unique radial

p
2 M
1
(M
p;q
) with radial part  2 M
1
(
q
),i.e.,

p
is U
p
invariant and'
p
(
p
) = :
(S
p
n
)
n
radial RW on M
p;q
with radial distribution  2 M
1
(
q
)
Problem:Limit theorems for'
p
(S
p
n
) for n;p!1
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 14/16
Theorem
Let  2 M
1
(
q
) such that
R

q
ktk
4
d(t) < 1.
If
n=p
n
!1;then
p
p
n
n

(S
p
n
n
)

S
p
n
n
nm
2

d
!N

0;T

If
n=p
n
!0;then
1
p
n

(S
p
n
n
)

S
p
n
n
nm
2

d
!N

0;

If
n=p
n
!c;then
1
p
n

(S
p
n
n
)

S
p
n
n
nm
2

d
!N

0;+cT

where
m
2
:=
Z

q
x
2
d(x) 2 
q
;
(T)
(i;j)(k;l )
:= (m
2
)
i;k
(m
2
)
j;l
+(m
2
)
i;l
(m
2
)
j;k
2 
q
2
:= Cov('
p
n
(
p
n
)) 2 
q
2
As in the case q = 1 we prove the theorem with the method of moments.
The essential ingredient is the calculation of the asymptotic behaviour of
the radial measure 
p
and their moments for p!1.
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 15/16
Proof:Let  2 N
pq
0
,k:= jj and R
i
():=
P
q
j=1

ij
.Then
CASE 1:If R
i
() is odd for some i then m

(
p
) = 0.
CASE 2:If R
i
() is even for all i then
m

(
p
) = C
X
2N
l
:jj=q
1
()


Z

q
D

Z
d

((zr)

(zr))
jz=0
d(r);
where
C:= C(k) > 0 is a constant,D

:=
@

11
@z

11
11

@

12
@z

12
12
  
@

pq
@z

pq
pq
()


is the generalized Pochhammer symbol:
()


:=
Y
q
j=1
( (j 1)=)

j
;:= dp=2 and := 1=d;
Z
d

is spherical polynomial of index  on 
q
= 
q
(F).
For F = R and C we have zonal polynomials Z
1

and Schur
polynomials Z
2

resp.
Thus,we calculate
m

(
p
) =
Z
M
p;q
x

11
x

12
   x

pq
d
p
(x) = O(p
jj=2
):
Waldemar Grundmann (TU Dortmund)
Limit Theorems
M

unchen,September 2012 16/16