From PET to SPLIT
Yuri Kifer
PET: Polynomial Ergodic Theorem (Bergelson)
in
T
fi
2
L
 preserving and weakly mixing
is
bounded measurable functions
qi
and
polynomials,
integer valued on the integers
SPLIT: sum product limit theorem
WANT TO PROVE that

1  N q1 ( n )
q (n)
f1  T
f    fi d  ) 
  (T
N  n 1
i 1

Is asymptotically normal: strong mixing conditions are needed
If
f i   Ai
N
T
n 1
q1 ( n )
then
f1 ( x) T
q (n)


qi ( k )
f ( x )  # k :
T
x  Ai 
 i 1

Application to multiple recurrence


setup:
• X 1 ,..., X ; X i  D  
bounded stationary
processes
•
if k` k and l  l`   algebras
•   mixing coefficient
•

approximation coefficient
• Assumption:
Functions q:
• q1 (n),..., q (n) nonnegative integer
valued on integers functions, satisfying
q1 (n)  rn  p for integer r  0, p  0

q j (n  1)  q j (n)  n ,   (0,1),n  n 0  1
1



q j 1 (  n
 )  q j ( n) n
Result:
Set
a j  EX j (0)
then
Is asymptotically normal with zero mean and the
T variance
h
e
where
when variance is zero?
 0
either
or
X j (0)  0
X j (0)  a j
almost surely for some
for all
j2
and
U unitary on , X random variable from
2
L (   X1, P)
j
Functional CLT
• For u  [0,1] set
u  [0,1]
Then in distribution
WN   W
W
Brownian motion
as
N 
Applications
Basic (splitting) inequalities
Y
and Z are
and
measurable, respectively, then
-
Let Y ( j ), j  0,1,...be bounded random variables
and
then for
where
variance:
• Set
Then
relying on estimates of
Some estimates
we
have
j1  j2
| k1  k2 |
and
Using splitting inequalities we get that
is small if j1  j2 or | k1  k2 | is large and
is small and the limit of the previous slide follows.
more delicate Gaussian estimates
• Let
and
.
Then
Block technique
• Set
and
Define for k  1, 2,..., m( N ) :
where
characteristic functions
• Set
and
then
Set
Then (main estimate):
Main estimate: idea of the proof
1st step:
2nd step:
3-d step:
• Then
4th step:
• Writing powers of sums as sums of products
the estimate of J (t , N ) comes down to the
estimate of
Next, we change the order of products in the
two expectations above so that the product 
appear immediately after the expectation and
apply the “splitting” inequality times to the
latter product for both expectations.
j1
we obtain
and
Next,
• For each fixed j we apply the “splitting estimate”
m( N )
to the product k 1 after the expectation and in
view of the size of gaps Z k between the
• blocks Yk we obtain
Collecting the above estimates the main estimate
follows.
Conclusion of proof:
• using
• and Gaussian type estimates above we
• obtain
• and, finally,
Concluding remark:
• the proof does not work when, for instance,
l  2, q1 (n)  n, q2 (n)  2n
Still, if X (0), X (1), X (2),...
are i.i.d. then
is asymptotically normal!
This can be proved applying the standard clt for triangular
1/ 2
W

N
series to
N
where
SN ( ) 

j:1 2 j  N

0 odd  N
SN ( )
( X (2 j ) X (2 j 1 )  EX 2 (0))