From PET to SPLIT Yuri Kifer. PET: Polynomial Ergodic Theorem (Bergelson) preserving and weakly...
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Transcript of From PET to SPLIT Yuri Kifer. PET: Polynomial Ergodic Theorem (Bergelson) preserving and weakly...
From PET to SPLIT
Yuri Kifer
PET: Polynomial Ergodic Theorem (Bergelson)
ifT preserving and weakly mixing is
bounded measurable functions
iq polynomials, integer valued on the integers
and
in 2L
SPLIT: sum product limit theorem
1 ( ) ( )1
1 1
1( )
Nq n q n
in i
T f T f f dN
ii Af
Is asymptotically normal: strong mixing conditions are needed
WANT TO PROVE that
If then
1 ( )( ) ( )1
1 1
( ) ( ) # : i
Nq kq n q n
in i
T f x T f x k T x A
Application to multiple recurrence
setup:
• bounded stationary processes
• algebras• mixing coefficient
• approximation coefficient
• Assumption:
1,..., ; iX X X D
if ` and ̀ k k l l
Functions q:
• nonnegative integer valued on integers functions, satisfying
1( ),..., ( )q n q n
1( ) for integer 0, 0q n rn p r p
0( 1) ( ) , (0,1), n n 1j jq n q n n
11( ) ( )j jq n q n n
Result:
Set then(0)j ja EX
Is asymptotically normal with zero mean and The
the variance
where
when variance is zero?
(0) 0jX
0 jeither almost surely for some
or (0)j jX a for all 2j and
unitary on , random variable fromU X
21( , )L X P
Functional CLT
• For set
[0,1]u
Then in distribution
NW W as N
Brownian motionW
[0,1]u
Applications
Basic (splitting) inequalities
and are and -measurable, respectively, then
Y Z
Let be bounded random variables and
( ), 0,1,...Y j j
then for
where
variance:
• Set
Then
relying on estimates of
Some estimates
1 2j j 1 2| |k k
1 2j j
we have
and
Using splitting inequalities we get thatis small if or is large and
is small and the limit of the previous slide follows.
1 2| |k k
more delicate Gaussian estimates
• Let .Then
and
Block technique
• Set and
Define for 1,2,..., ( ) :k 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 comes down to the estimate of
( , )J t N
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 .
1j
we obtain
and
Next,
• For each fixed we apply the “splitting estimate” to the product after the expectation and in view of the size of gaps between the
• blocks we obtain
j
Collecting the above estimates the main estimate follows.
( )
1
m N
kkZ
kY
Conclusion of proof:• using
• and Gaussian type estimates above we
• obtain
• and, finally,
Concluding remark:• the proof does not work when, for instance,
1 22, ( ) , ( ) 2l q n n q n n Still, if (0), (1), (2),...X X X are i.i.d. then
is asymptotically normal!
This can be proved applying the standard clt for triangular
series to
where1 2
:1 2
( ) ( (2 ) (2 ) (0))j
j jN
j N
S X X EX
1/ 2
0
( )N Nodd N
W N S