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