EXPONENTIAL CHI SQUARED DISTRIBUTIONS IN INFINITE … · 2016. 7. 19. · EXPONENTIAL CHI SQUARED...

EXPONENTIAL CHI SQUARED DISTRIBUTIONS ININFINITE ERGODIC THEORY

JON. AARONSON & OMRI SARIG

Abstract. We prove distributional limit theorems for randomwalk adic transformations obtaining ergodic distributional limitsof exponential chi squared form.

§0 Introduction

As in [A1], for X,B,m a σ-finite measure space, Fn X 0,ªmeasurable, and Y > 0,ª a random variable, we say that Fn con-

verges strongly in distribution to Y , (written FndÐnª

Y,) if it converges

in law with respect to all m-absolutely continuous probabilities; equiv-alently

gFn Ðnª

EgY weak in Lªmfor each bounded, continuous function f 0,ª R.

For discussion of strong distributional convergence, see [A1], [A2],[E] and [TZ].

Here, we study distributional stability. As in [A2], we’ll call aconservative, ergodic measure preserving transformations X,B,m,T distributionally stable if there are constants an A 0 and a randomvariable Y on 0,ª so that

1anSnf d

Ð Y mf ¦ f > L1

where Snf Pn1k0 f X T

k and mf RX fdm.By the ratio ergodic theorem, if the above convergence holds for some

f > L1, then it holds ¦ f > L1

.

2010 Mathematics Subject Classification. 37A40, 60F05, (37A05, 37A20, 37A30,37B10, 37D35).

Key words and phrases. Infinite ergodic theory, distributional convergence, ran-dom walk adic transformation.

Aaronson’s research was supported by Israel Science Foundation grant No.1114/08. Sarig’s research was supported by the European Research Council, grant239885.

1

2 Jon. Aaronson & Omri Sarig ©2012

If the ergodic distributional limit Y is integrable and normalized byEY 1, the constants an are determined uniquely up to asymptoticequality and are also known as the return sequence of T . Both the(normalized) ergodic distributional limit and the return sequence areinvariant under similarity (see [A1]).

By the Darling-Kac theorem ([DK]), pointwise dual ergodic transfor-mations (e.g. Markov shifts) with regularly varying return sequencesare distributionally stable with Mittag-Leffler ergodic distributionallimits (see also [A1], [A2]).

Our present study concerns random walk adic transformations.

A random walk adic transformation is a conservative, ergodic mea-sure preserving transformation associated to a Markov driven, aperi-odic, random walk on a group of form G Zk RDk. These were firstconsidered in [HIK] (and are sometimes known as ”generalized HIKtransformations”).

It is the (unique) G-extension of the adic transformation on theunderlying Markov shift which parametrizes the tail relation of therandom walk. This definition is explained in §1.

The degree of a random walk adic transformation is the dimensionof the associated group G: dimspanG. It appears as the number ofdegrees of freedom in the χ2 distribution appearing in the limit.

We establish the following

TheoremSuppose that X,B,m,T is a random walk adic transformation with

degree D > N, then

1anT Snf d

Ð 2D2 e

12χ2Dµf ¦ f > L1

where Snf Pn1k0 f X T

k, anT nlognD~2 ; and χ2D YξY2

2 for ξ a

standard Gaussian random vector on RD.

Here, for an, bn A 0, an bn means § limnªanbn > 0,ª.

This ergodic distributional limit first appears in [LS] (see below).Most of the paper is devoted to proving the theorem. In §1, we

define adic transformations and random walk adic transformations asgroup extensions of adic transformations. In §2, we establish compactrepresentation for (all) adic transformations and a uniform convergencetheorem for stationary adic transformations which latter is needed inthe proof of the theorem. We review the distributional limit theory of

Exponential χ2 distributions 3

Markov shifts in §3 and prove the theorem in §4, giving applications toexchangeability in §5 and horocycle flows in §6.

Related, earlier work can be found in [AW], [ANSS] and [LS] (see§6).

§1 Bratteli diagrams, adic and random walk adictransformations

Bratteli diagrams.Fix bn C 2 n C 1 and set Sk 0,1, . . . , bk 1, Ω Lª

k1 Sk.A Bratteli diagram is a subset Σ ` Ω of form

Σ ω > Ω Akωk, ωk1 1 ¦ k C 1where for k C 1, Ak Sk Sk1 0,1 is the kth transition matrix.

We assume throughout that the transition matrices are non-degeneratein that they have no zero rows or columns.

Recall that Ω is compact when equipped with the standard metricdx, y expminn xn x yn and Σ is a closed subset.

The Bratteli diagram Σ is called stationary if Sk S, Ak A ¦ k C 1.In this case, Σ is a topological Markov shift (TMS) with transtitionmatrix A as in [LM].

The only result in this paper concerning non-stationary Bratteli di-agrams is the compact representation lemma in §2.

Tail relation on a Bratteli diagram.The tail relation on Σ is the equivalence relation

T TΣ x, y > Σ Σ §n such that xªn yªn where xªn xn, xn1, . . . .

The equivalence classes of the tail relation are linearly ordered by thereverse lexicographic order, namely the partial order h on Σ defined by

x h y §n s.t. xªn1 yªn1 and xn @ yn.

If x is maximal, then xn maxy > Sn Any, xn1 1 ¦ n C 1;therefore the collection of non-maximal points is open and the collectionΣmax of maximal points is closed. A similar argument shows that theset Σmin of minimal points is closed.

In case Σ is a TMS (stationary Bratteli diagram), more is true.


If x is maximal, then xn ϕxn1 where ϕx maxy > S

Ay, x 1 (well defined by non-degeneracy); and we claim that

x is periodic, with period B #S.(R)

To see (R), note first that § s > S so that #n C 1 xn s ª.The sequence n( ϕns is eventually periodic with a final period

t, ϕt . . . , ϕκ1 t ϕJs, . . . , ϕJκ1

swith J C 1 & κ B #S.

Let p ϕκ1 t, ϕκ2

t, . . . , t, ϕκ1 t, ϕκ2

t, . . . , t, . . . . We proveour claim by showing that x σ`p for some ` C 1. Let nk ª be sothat xnk s ¦ k C 1. It follows that xnkν ϕ

νs whence § 1 B `k B κ

so that xnkJ1 σ`kpnkJ1 . There is a subsequence mj nkj ª so

that `kj `k1 ` ¦ j C 1 whence x σ`p and (R) is established.Thus the set of maximal points (and the set of minimal points) is

finite.

Adic transformations.The adic transformation (generated by h) on the Bratteli diagram Σ

is

τ Σ Σmax Σ Σmin defined by τx miny y i x.It is called stationary if the underlying Bratteli diagram is stationary.

As shown in [V], every ergodic, probability preserving transformationis isomorphic to some adic transformation.

Stationary adic transformations areY isomorphic to odometers or primitive substitutions, have zero en-tropy but can be weakly mixing (see [L1]); andY are always uniquely ergodic; moreover the unique τ–invariant proba-bility measure ν0 is globally supported, non-atomic, Markov and equiv-alent to the Parry measure µ (of maximal entropy) for the associatedTMS (see [BM]).

The Σ, f-random walk.Let Σ be a topologically mixing TMS on the (ordered) finite state

space S, let σ Σ Σ be the shift and let τ Σ Σ be the corre-sponding stationary adic transformation where

Σ Σ

nC0

σnΣmax 8Σmin n>Z

τnΣ Σmax 8Σmin.The tail relation of Σ is the tail relation of σ:

TΣ Tσ nC0

x, y > Σ Σ σnx σny;


and this is parametrized by the adic transformation:

Tσ 9 ΣΣ x, τnx x > Σ, n > Z.

A function f Σ Rd is Holder continuous if § θ > 0,1, M A 1 sothat

Yfx fyY BMθn ¦ x, y > Σ, xªn yªn .(G)

Specifically for we call f Σ Rd θ-Holder continuous (θ > 0,1) if(G) is satisfied for some M A 1.

For f Σ Rd Holder continuous let

H `fnx n C 1, x > Σ, σnx xewhere `e means “the group generated by” and

fnx f σn x n1

Qk0

fσkx.Let

G `fnx fny n C 1, x, y > Σ, σnx x, σny ye,then G, H are both closed subgroups of Rd and G B H.

It follows from Livsic’s cohomology theorem [L2], (see e.g. [ANS],[SA], [PS]) that

f g g X σ h a where

Y g Σ Rd is Holder continuous;

Y a > H is such that `G ae H;

Y h Σ G is Holder continuous and σ-aperiodic in the sense thatif γ > ÂG, λ > S1, g Σ S1 Holder continuous and γ X f λgXσg , thenγ 1 λ 1 and g is constant.

It follows that dimG dimH D where for A ` Rd, dimAdenotes the dimension of the closed linear subspace spanned by A.

Any closed subgroup G B RD with dimG D is conjugate bylinear map to a group of form Zk RDk where 0 B k B D dimGand Z0, R0 0.

Now suppose that f Σ G ZkRDk is Holder continuous and σ-aperiodic and consider the Σ, f-random walk ΣG,BΣG, Çm,σfwhere σf Σ G Σ G is defined by

σfx, y σx, y fx & dÇmx, y dµxdywhere µ is the σ-invariant Parry measure (with maximal entropy) anddy is Haar measure on G.


We note for future reference that σnf x, y σnx, y fnx.As shown in [G], by the aperiodicity of f , Σ G, σf , Çm is exact.

Random walk adic transformation over Σ, f, τ.The random walk adic transformation over Σ, f, τ is that skew

product over τ which parametrizes the tail Tσf of the Σ, f-randomwalk. To see existence and uniqueness of such:

Tσf nC0

x, y, x, y > Σ G2 σnf x, y σnf x, y

nC0

x, y, x, y > Σ G2 σnx σnx & y fnx y fnx

x, y, x, y > Σ G2 x,x > Tσ & y y ψx,x

where

ψx,x ª

Qk0

fσkx fσkx.Thus

Tσf 9 ΣG2 x, y, x, y > Σ

G2 σnf x, y σnf x, y

x, y, T nx, y x, y > ΣG2, n > Z

holds for T Σ G Σ G of form

T x, y τφx, y τx, y φxif and only if

φx ψx, τx ª

Qk0

fσkτx fσkx.We consider T with the invariant measure

dmx, y dνxdywhere ν > PΣ is the τ–invariant Markov measure (equivalent to theParry measure µ) and dy is Haar measure on G.

As mentioned above, it was shown in [G] that Σ G, σf , Çm is ex-act, whence Σ G, T,m is ergodic and also conservative (being aninvertible, ergodic, measure preserving transformation of a non-atomicspace).

The degree of the random walk adic transformation Σ G, T,m is

deg T dim spanG.In this paper we ignore the other invariant measures for T (which

are considered in [ANSS]).


§2 Uniform Convergence

Uniform Convergence LemmaLet Σ be a mixing TMS and let τ Σ Σ be the associated sta-

tionary adic transformation with τ–invariant Borel probability measureν0 > PΣ, then

1

n

n1

Qk0

F X τ k ÐÐnª

SΣ

Fdν0 uniformly on Σ ¦ F > CΣ.Although τ is a uniquely ergodic homeomorphism on Σ, this space isnot compact.

The main part of the proof is to provide a suitable continuous trans-formation of a related compact space. This latter construction is madefor any adic transformation.

Compact Representation LemmaLet Σ be a Bratteli diagram and let τ be the associated adic transfor-

mation. There are:Y a compact metric space ÂΣ, Âd;

Y a continuous injection π Σ Σmax ÂΣ and

Y continuous surjections $ ÂΣ Σ, Âτ ÂΣ ÂΣ so that

$ X π IdSΣΣmax , π X τ Âτ X π & $ X Âτ τ X$.Proof. For ω > Σmax, set

Aω α > Σ § sequence xn > ΣΣmax s.t. xn ω and τxn α,then Aω ` Σmin.

Let

Σ0 Σ Σmax & Σ1 ω>Σmax

ω Aω.Define the metric space ÂΣ, Âd by

ÂΣ Σ0 >Σ1; ÂdSΣ0Σ0 d & for ω,α > Σ1

Âdω,α, z dω, z dα, τz z > Σ0;

dω,ω dα,α z ω, α > Σ1.

To see that this is a compact metric space, let znnC1 be a sequence inÂΣ, then:


Y If § nk ª, znk ωk, αk > Σ1, then (possibly passing to asubsequence) we may assume that ωk, αk ω,α > Σmax

Σmin. To see that ω,α > Σ1, for each k C 1, choose xk > Σ0

so that dxk, ωk dτxk, αk 0. It follows that ω,α > Σ1

and ωk, αk ω,α in ÂΣ.Y Otherwise § nk ª, znk > Σ0, znk r > Σ and

– if r > Σ0, then znk r in ÂΣ;– if r > Σ1, then § m` nk` ª so that τznk s > Σ;

whence r, s > Σ1 and zm` r, s in ÂΣ.

Let π Σ0 ÂΣ be the identity map. The following map is a contin-

uous left inverse: $ ÂΣ Σ defined by the identity map on Σ0 and byω,α( ω on Σ1.

Now define Âτ ÂΣ ÂΣ by Âτx τx for x > Σ0 and Âτω,α α forω,α > Σ1, then Âτ is continuous and π X τ Âτ X π.

To see that Âτ is onto, it suffices to show that ÂτÂΣ a Σmin. To this end,fix α > Σmin, then § xn > Σ Σmin, xn α. Without loss of generality,xn ¶ Σmax and so xn τyn for some yn > Σ where yn ω > Σ. It

follows that ω > Σmax (else α τω ¶ Σmin) whence ω,α > ÂΣ and $is onto.

Proof of the Uniform Convergence LemmaSince ν0 > PΣ, it lifts to a Âτ–invariant measure ν1 > PÂΣ: ν1Xπ1

ν0. We claim that Âτ is uniquely ergodic on ÂΣ with invariant measureν1. Else § ν1 x ν2 > PÂΣ with ν2 X Âτ1 ν2. This entails ν2 X$1 ν0

whence ν2 ν1.It follows that

∆nF supÂΣW 1n

n1

Qk0

F X Âτ k SÂΣFdν1WÐÐnª

0 ¦ F > CÂΣ.If f > CΣ, then f X$ > CÂΣ and for every ω > Σ and k C 1, we haveτ kω > Σ whence fτ kω f X$Âτ kπω. Thus

supω>Σ

T 1n

n1

Qk0

fτ kω SÂΣ fdν1T B ∆nf X φÐÐnª

0. V

§3 Limit theory for the shift

Let Σ,A, µ, σ be a mixing TMS with µ a σ-invariant, Markov mea-sure let Âσ Âσµ be its transfer operator, let G ` Rd be a closed subgroup


of dimension D; and let f f 1, . . . , f D Σ G be Holder contin-uous and σ-aperiodic. Let f f Ef. Note that f may have valuesoutside G.

We’ll need the following results for the sequel. The results are notnew although we did not find references for their statements. Theproofs are standard and will only be sketched.

3.1 Asymptotic variance theorem§ a D D, symmetric, positive definite matrix Γ Γf so that

RankΓf D &1

nEf in f

jn Ð

nªΓi,j ¦ 1 B i, j BD

where fin Pn1

k0 fiX σk.

Since Γ is symmetric and positive definite, it can be put in the form

Γ U tMU tMtwith U unitary and M A 0 diagonal.

3.2 Central limit theorem

(CLT) Âσn1 fnºn>I Ðnª

1»2πD det ΓSI

exp12u

tΓ1uduwhenever I ` RD is Riemann integrable (i.e. with Riemann integrableindicator function).

3.3 Local limit theorem (mixed lattice-nonlattice case)

Let 0 > I ` G be Riemann integrable; then(LLT)

ºnDÂσn1fn>nEfºnxnI ÐÐÐÐÐÐnª, xnu

mGI»2πD det Γ exp1

2utΓ1u.

Proof sketchesSuppose that f is θ-Holder continuous and let ρ ρθ be the metric

on Σ defined by

ρx, y θinf nC1 xnxyn.Let L Lθ be the Banach space of ρ-Lipschitz continuous (equivalently,θ-Holder continuous functions) on Σ equipped with the norm

YF YL supx>Σ

SF xS supx, y>Σ

SF x F xSρx, y .


As shown in [D-F], [GH], for some N C 1, ÂσN L L satisfies theDoeblin-Fortet inequality, namely § r > 0,1 & H A 0 so that

(DF) YÂσNF YL B rYF YL H supx>Σ

SF xS ¦ F > L,

whence ([D-F], [IT-M]) Âσ L L is quasi-compact in that § θ >0,1 & M A 1 so that

(QC) YÂσnF SΣFdµYL BMθnYF YL ¦ F > L.

Proof of the asymptotic variance theoremThe absolute convergence of the series PnC1 S RΣ f i f j X σndµS for

1 B i, j B D follows from (QC) and the convergence follows from this.The non-singularity of the limit matrix follows from the aperiodicity ofeach t f Σ R t > RD via Leonov’s theorem (as in [RE]). V

Proof of the central and local limit theoremsBoth of these results are established using Nagaev’s perturbation

method as in [HH], [PP], [RE] (also known as characteristic functionoperators [AD]).

For t > CD define Pt L1Σ L1Σ by PtF ÂσeitfF , thenPtF L L and the map t( Pt is analytic CD homL,L with

∂r

∂tk1∂tkrPtF irÂσ r

Mj1

f kjeitfF ;where homL,L is equipped with the uniform topology. We have (see[N], [PP] &/or [RE])

Y YPtY B 1 ¦ t > ÂG Tk RDk with equality iff t 0;

Y Pt satisfies (DF) for StS small; whence

Nagaev’s Theorem [N] There are constants ε A 0, K A 0 andθ > 0,1; and analytic functions λ ε, ε BC0,1, N ε, ε homL,L such that

YP nt h λtnNthYL BKθnYhYL ¦ StS @ ε, n C 1, h > L

where ¦StS @ ε, Nt is a projection onto a one-dimensional subspace,λ0 1 & N0F RΣFdµ.

The expansion of λ is obtained by considering t f Σ R t > RDas in [GH]. It is given by

λt 1 it Ef @ Γt, t A

2 oStS2 as t 0.


The central limit theorem follows from this in the standard manner(see [GH], [RE]); and the local limit theorem follows with a proof as in[S] (see [AD]).

§4 Proof of the theorem

For n C 1, set `n logλ n where λ ehtopΣ. Let 0 > I ` G beRiemann integrable with 0 @ SI S @ ª of form I 0k J where

0k > Zk, 0kj 0 1 B j B k and 0 > J ` RDk is Riemann integrable

with 0 @ SJ S @ª. Here, S S denotes Haar measure on G.To achieve our goal, we’ll establish:

¦ R A 0, ¦ x > Σ,(o)

`D~2n

n Sn1ΣIx,01BRf`nxº

`n

SI S»2πD det Γ exp YM1Uf`nxY2

2`n1BRf`nxº

`n

where f f Ef and BR z > RD YzY @ R and an bn meansan bn Ð

nª0.

We’ll show first that (o) holds, and then we’ll prove that (o) Ôthe theorem.

Overview of the proof of (o).The proof uses a process of block splitting where in order to

estimate

Sn1ΣIx,0 n1

Qj0

1ΣIτ kx,φkxwe split the τ -orbit block τ kx 0 B k B n 1 into simpler blocks onwhich it is easy to apply the results of §3.

This is done as follows.For x > Σ, N C 1

σNσNx τ kxmin 0 B k B #σNσNx 1where xmin min σNσNx with respect to the reverse lexicographicorder and

#σNσNx1

Qj0

1ΣIXTkxmin,0 #y > σNσNx fNy > fNxminI.


Quantities appearing, such as

#y > σNx fNy > fNz Iwhere I ` G is Riemann integrable, are estimated using (LLT) as inlemma 4.1 (below).

The arbitrary blocks are estimated from the simple ones of suitablysmaller size. This is calibrated using the Perron-Frobenius theorem:

#σNσNx Q0BsBd1

ANs,xN1 cxN1λN

where A is transition matrix of Σ, λ ehtopΣ is its leading eigenvalueand c S 0,ª.

Fix M C 1 large. For each n C 1 large, let N Nn be such thatMλN λ1n, then

τ jx 0 B j @ n M1

#k0

σNτ kσNxup to relatively small edge effects (estimated in the proof below usinglemma 4.2) and

Sn1ΣIx,0 M1

Qk0

#y > σNτ kσNx fNy fNmin σNτ kσNxup to relatively small errors (estimated in lemma 4.3 below).

Proof of (o) on page 11.

For x > Σ, tn > G, supnYtnYº

n@ª, set

Nnx #z > σnσnx fnz > tn I.Lemma 4.1

Nnx λnhσnxSI S»2πnD det Γ

expYM1UtnY2

2n

uniformly on Σ where h dµdm ; m and µ being the τ - and σ-invariant

probabilities (respectively).

Proof

Let Âσm, Âσµ be the transfer operators of σ with respect to m & µ

respectively, then Âσmf hÂσµfh and


Nnx λnÂσnm1fn>tnIσnx λnhσnxÂσnµ1fn>tnI

h σnx

λnhσnxSI S»2πnD det Γexp 1

2nYM1UtnY2

uniformly on Σ by (LLT). V

Block splitting.For n C 1, let Σn,s x1, . . . , xn x > Σ, xn1 s, Jns #Σn,s,

then

Jns Qu>S

Anu,s nª

csλnuniformly in s > S where λ ehtop.Σ,σ and c S R.

It will be convenient also to set ÂJnz #σnz. Here

ÂJnx #Σn,x1 Jnx1and ÂJnx cxλnuniformly in x > Σ where c Σ R, cx cx1.

For n C 1 fixed, we call a point x > Σ

Y n-minimal if x min σnσnx miny > Σ yn1 xn1 &

Y n-maximal if x max σnσnx maxy > Σ yn1 xn1;Now define

Kn Σ N & τn Σ Σ by

Knx mink C 1 τ kx is n-maximal & τnx τKnx1,

then:

Y σnτnx τσnx;Y τnx is n-minimal and

Y Knx B ÂJnσnx #σnσnx with equality if x is n-minimal.

It follows that for j C 1,

σnτ jnx τ jσnxand

Knτ jnx ÂJnτ jσnx.


For n C 1 fixed and r C 1, set

Krn x r1

Qj0

Knτ jnx Knx r1

Qj1

ÂJnτ jσnx.Lemma 4.2 § ηn, θr 0 so that

Krn x eηnθrrλnEc ¦ n, r C 1, x > Σ.

Proof By the uniform convergence lemma § θr 0 such that

r1

Qj1

cτ jσnx eθrrEc ¦ x > Σ, n, r C 1.

Suppose that Jns eηnλncs where ηn 0, then for x > Σ,

Krn x Knx r1

Qj1

ÂJnτ jσnxKnx eηnλn r1

Qj1

cτ jσnxKnx eηneθrrλnEc.

Since Knx B ÂJnσnx Oλn, the lemma follows. V

Lemma 4.3 For r > N fixed, x > Σ and R A 0, as nª:

SKrn x1ΣIx,01BRfnxº

n à(1)

hr1σnxSI SλnnD~2 exp YM1UfnxY2

2n 1BRfnxºn

and

SKrn x1ΣIx,01BRfnxº

n ß(2)

hr1σnxSI SλnnD~2 exp YM1UfnxY2

2n 1BRfnxºn

where hrz Pr1j0 hτ jz.

Here, for An, Bn A 0, An à Bn means limnªAnBn

C 1 and An ß Bn

means limnªAnBn

B 1

Proof


Writing K0n 0, we have

SKrn x1ΣIx,0

r1

Qj0

SKj1n x1ΣIx,0 SKj

n x1ΣIx,0 SKnx1ΣIx,0

r1

Qj1

SKj1n x1ΣIx,0 SKj

n x1ΣIx,0 .For fixed j C 1,

SKj1n x1ΣIx,0 SKj

n x1ΣIx,0

Kj1n x1

QkK

jn x

1ΣIτ kx,φkx

Knτ jnx1

Q0

1ΣIτ `τKjn xx, φ

Kjn x`x

ÂJnτ jσnx1

Q0

1ΣIτ `τ jnx, φKjn x`x

Now,

τ `τ jnx 0 B ` B ÂJnτ jσnx 1 σnτ jσnx,φKjn x`x ψz, x

ª

Qk0

fσkx fσkz 0 B ` B ÂJnτ jσnx 1;and so

SKj1n x1ΣIx,0 SKj

n x1ΣIx,0 #z > σnσnτ jnx ψz, x > I.

For z τ `τ jnx > σnτ jσnx we have

φKjn x`x ψz, x

ª

Qk0

fσkx fσkz ftx ftz

where t tx, z minN C 1 σNx σNz.Here, σnz τ jσnx and

tx, z B n tτ jσnx,σnx.Thus


ψz, x fnz fnx κn,jxwhere

Sκn,jzS B 2 sup Sf SN n B 2 sup Sf Stτ jσnx,σnx.We claim that for a.e. x > Σ,

max1BjBr

tτ jσnx,σnx Ologn as nª.(M)

Proof of (M)For M A 0 set AnM x > Σ tτσnx,σnx AM logn, then

mx > Σ max1BjBr

tτ jσnx,σnx AM logn Bm 0BjBr1

τjAnMB rmAnM.

Now tτσnx,σnx AM logn iff § z > Σmax so that

xnj zj ¦ 1 B j BM logn.

Thus

mAnM B sup hµAnM sup h Q

z>Σmax

µz1, . . . , zM logn OλM logn

and PnC1mAnM @ ª for M A 1logλ . The claim (M) now follows

from the Borel-Cantelli lemma. V

In view of (M), we have by lemma 4.1 that for a.e. x:

1BRfnxºnS

Kj1n x1ΣIx,0 SKj

n x1ΣIx,0 1BRfnxº

n#z > σnσnτ jnx fnz > fnx κn,jx I

SI S1BRfnxºnλnhσnx

nD~2 exp YM1Ufnxκn,jxY22n

SI S1BRfnxºnλnhσnx

nD~2 exp YM1UfnxY22n .

The lemma follows from this. V

Completion of the proof of (o) on page 11Given 0 @ ε @ 1

3 ,


Y use the uniform convergence lemma to fix rε such that ¦ y > Σ, r Crε, eθr @ 1 ε where θr is as in lemma 4.2, and

cry, cr2y 1 εrEmc & hry, hr2y 1 εrEmh.Y fix J A eθr ¦ r C 1 and for n C 1 let

Ln Ln,ε logλn

2JEcrε and let rn rn,ε be such that

KrnLn

τσnx B n @Krn1Ln

τσnx @Krn2Ln

σnx.It follows that

SKrnLn

τσnx1ΣIx,0 B Sn1ΣIx,0 B SKrn2Ln

σnx1ΣIx,0.By lemma 4.2,

n BKrn2Ln

σnx B eηLnθrnrnλLnEc ß eθrnrn n2JEcrεEc B nrn

2rε

whence for large n, rn A rε and

Sn1ΣIx,01BRfLnxºLn

C SKrnLn

τσnx1ΣIx,01BRfLnxºLn

à

hrnτσLnxλLnLn

D~2 exp YM1UfLnxY22Ln

1BRfLnxºLn

C 1 εSI SλLnrn

LnD~2 exp YM1UfLnxY2

2Ln1BRfLnxº

Ln

and similarly

Sn1ΣIx,01BRfLnxºLn

B SKrn2Ln

σnx1ΣIx,01BRfLnxºLn

ß

hrn2σLnxλLnLn

D~2 exp YM1UfLnxY22Ln

1BRfLnxºLn

B 1 εSI SλLnrn

LnD~2 exp YM1UfLnxY2

2Ln1BRfLnxº

Ln.

Now,

Y Ln `n logλ n ¦ ε A 0 and since rn A rε,

n CKrnLn

τσnx C eηnθrnrnλLnEc à 11εrnλ

LnEc.n @K

rn1Ln

τσnx ß 1 εrnλLnEc @ 11εrnλ

LnEcwhence

λLnrnLn

D~2 1 ε2 n

`D~2n Ec .

This proves (o) (on page 11). V


Proof that (o) Ô the theorem. Let Γ be as in §3 and writeΓ V V t where V U tM with U unitary and M A 0 diagonal.

Let ξ ξ1, . . . , ξd where ξ1, . . . , ξd are independent, identically dis-tributed, Gaussian random variables with Eξj 0, Eξ2

j 1, thenZ U tMξ V ξ is Gaussian with correlation matrix

EZiZj EQs,t

Vi,sξsVj,tξt Qs

Vi,sVj,s Γi,j.

By (CLT)

fnºn

dÐ U tMξ Z.

Now suppose that (o) (as on page 11) holds. We’ll show that for somean n

`D~2n

and for g > C0,ª, f > L1m,

g 1an Snf Ðnª Ege 1

2χD2 mf weak in Lªm.

By the asymptotic variance theorem, EYfnY2 Oºn and ¦ ε A

0 § R so that mΣCnR A 1 ε ¦ n C 1 where CnR f`nxº`n

>

BR.Thus for n > N & R A 0 both large enough and x > CnR we have

g `D~2n

n Sn1ΣIx,0 g SI S»2πD det Γ exp YM1UfnxY2

2n ε.Next, by (CLT),

SΣI

g SI S»2πD det Γ exp YM1UfnY2

2n dm Ðnª

Eg SI S»2πD det Γ exp YM1UZY2

2 Eg SI S»2πD det Γ

expχ2D

2 .Thus, § an n

`D~2n

,

SΣI

g 1an Sn1ΣIdm Ð

nªEgmGI 2D~2e 1

2χ2D.

Using Corollary 3.6.2 of [A1], we obtain that ¦ F > L1m, g >

C0,ª,g 1

an SnF Ðnª

EgmF 2D~2e 12χ2D weak in Lªm

where mF RΣGFdm. V


§5 Application to exchangeability

Let S 0,1, . . . , d 1 and let Σ ` SN be a mixing TMS. DefineF Σ Zd1 by F xk δx1,k.

As shown in [ADSZ], F Σ Zd1 is σ-aperiodic iff Σ is almost ontoin the sense that¦ b, c > S, § n C 1, b a0, a1, . . . , an c > S such that

σak 9 σak1 x g 0 B k B n 1.Define ϕ Σ N and R Σ Σ by

ϕx minn C 1 τnxi xσi some finite permutation σ of Nand

Rx τϕxx.Corollary 5.1

Suppose that Σ is almost onto, then Σ,BΣ,R,m is an ergodic,probability preserving transformation and § bn nlognd1~2 such that

1

bnn1

Qk0

ϕ XRk dÐ e

12χ2d1 .

Proof.The random walk adic

Σ Zd1,BΣ Zd1,m mZd1 , T over Σ, F , τ is conservative and ergodic. Calculation shows thatTΣ0x,0 Rx,0 whence Σ,BΣ,R,m is an ergodic, probabilitypreserving transformation.

By the theorem,

1

anSnf dÐ e

12χ2d1µf ¦ f > L1

where an nlognd1~2 (we absorbed the factor in an).In particular,

1

anSn1Σ0 dÐ e

12χ2d1

whence by inversion (proposition 1 in [A2]),

1

bnn1

Qk0

ϕ XRk dÐ e

12χ2d1


where bn a1n nlognd1~2 . V

§6 Chi squared laws for horocycle flows

Let M0 be a compact, connected, orientable, smooth, Riemanniansurface with negative sectional curvature, and let T 1M0 denote the setof unit tangent vectors to M0. The geodesic flow on T 1M0 is the flowwhich moves a vector v > T 1M at unit speed along its geodesic.

Margulis [Mrg] and Marcus [Mrc] constructed a continuous flow ht T 1M T 1M such that

(a) The h–orbit of Ñv > T 1M0 equals

W ssÑv Ñu S distgsÑv, gsÑuÐÐsª

0(b) §µ s.t. gs X ht X gs hµ

st

In the special case when M0 is a hyperbolic surface, h is the stable horo-cycle flow. Properties (a) and (b) should be compared to the relationbetween the odometer and the left shift.

A ZD–cover of M0 is a surface M together with a continuous mapp M M0 such that p is a local isometry at every point, the group ofdeck transformations

G A M M SD an isometry s.t. p XA pis isomorphic to ZD, and for every x >M0, p1x is a G–orbit of somepoint in M .

The flows g, h T 1M0 T 1M0 lift to flows g, h T 1M T 1M whichcommute with the elements of G, and which satisfy (a),(b). Now (a)and (b) could be compared to the relation between the HIK transfor-mation and a ZD–skew-product over the left shift map [Po].

The locally finite ergodic invariant measures for h are described in[BL] and [S]. There are infinitely many, but only one up to normal-ization, is non-squashable [LS]. This measure, which we call m0, isrationally ergodic, and it is invariant under the action of the geodesicflow and the deck transformations.

We choose a normalization for m0 as follows. Let ÈM0 be a connectedpre-compact subset of M s.t. p ÈM0 M0 is one-to-one and onto, thenwe normalize m0 so that m0T 1ÈM0 1.

The following can be extracted from [LS]:


Theorem 6.1There exists aT T ~lnT D~2 such that for every f > L1m0 with

positive integral,

1

aT ST

0fhsωds d

ÐÐÐTª

2D2 e

12χ2Dm0f.

Proof sketchEnumerate G Aξ ξ > ZD such that Aξ

1X Aξ

2 Aξ

1ξ

2, then

M "ξ>ZD AξÈM0. The ZD–coordinate of Ñv > T 1M is the unique

ξÑv > ZD such that Ñv > T 1AξÈM0.It is known that 1º

Tξ X gT

dÐÐÐTª

N , where N is a D–dimensional

Gaussian random variable with positive definite covariance matrix CovN (Ratner [R], Katsuda & Sunada [KS]).

Let Y YH denote the norm on RD given by YvYH »vtCovN 1v.

The following is proved in [LS] (Theorem 5): Suppose f > L1m0, thenfor every ε A 0, for m0–a.e. Ñv > T 1M , for all T large enough

2D2εe

121ε] ξglogµ T Ñv¼

logµ T]2H B

1

aT ST

0fhsÑvds B 2

D2εe

121ε] ξglogµ T Ñv¼

logµ T]2H

where aT constT ~lnT D~2 (the value of the constant is known,see [LS]).

This is the version of (o) (see page 11) needed to deduce the theoremas above.

References

[A1] Aaronson, J. An introduction to infinite ergodic theory. Mathematical Sur-veys and Monographs, 50. American Mathematical Society, Providence,RI, 1997.

[A2] J. Aaronson, The asymptotic distributional behaviour of transformationspreserving infinite measures. J. Anal. Math. 39 (1981), 203-234.

[AD] Aaronson, Jon ; Denker, Manfred . Local limit theorems for partial sumsof stationary sequences generated by Gibbs-Markov maps. Stoch. Dyn. 1(2001), no. 2, 193–237.

[ADSZ] Aaronson, J.; Denker, M.; Sarig, O.; Zweimuller, R. Aperiodicity of co-cycles and conditional local limit theorems. Stoch. Dyn. 4 (2004), no. 1,31–62.

[ANSS] Aaronson, Jon; Nakada, Hitoshi; Sarig, Omri; Solomyak, Rita: Invariantmeasures and asymptotics for some skew products. Israel J. Math. 128(2002), 93–134. Corrections: Israel J. Math. 138 (2003), 377-379.

[ANS] Aaronson, J. ; Nakada, H. ; Sarig, O. Exchangeable measures for subshifts.Ann. Inst. H. Poincar Probab. Statist. 42 (2006), no. 6, 727–751.


[AW] J. Aaronson and B. Weiss, On the asymptotics of a 1-parameter family ofinfinite measure preserving transformations, Bol. Soc. Brasil. Mat. (N.S.),29, (1998), 181–193.

[BL] Babillot, M.; Ledrappier, F.: Geodesic paths and horocycle flow on abeliancovers. Lie groups and ergodic theory (Mumbai, 1996), 1–32, Tata Inst.Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998.

[BM] Bowen, Rufus; Marcus, Brian: Unique ergodicity for horocycle foliations.Israel J. Math. 26 (1977), no. 1, 43–67.

[D-F] Doeblin, W., Fortet, R. Sur des chaines a liaison completes Bull. Soc.Math. de France 65 (1937) 132-148.

[DK] D. A. Darling, M. Kac, On occupation times for Markoff processes, Trans.Amer. Math. Soc. 84, (1957), 444–458.

[E] Eagleson, G. K.: Some simple conditions for limit theorems to be mixing.(Russian) Teor. Verojatnost. i Primenen. 21 (1976), no. 3, 653–660. Engl.Transl.: Theor. Probability Appl. 21 (1976), no. 3, 637–642 (1977).

[G] Guivarc’h, Y. Proprietes ergodiques, en mesure infinie, de certainssystemes dynamiques fibres. Ergodic Theory Dynam. Systems 9 (1989),no. 3, 433–453.

[GH] Guivarc’h, Y. ; Hardy, J. Theoremes limites pour une classe de chaınesde Markov et applications aux diffeomorphismes d’Anosov. Ann. Inst. H.Poincare Probab. Statist. 24 (1988), no. 1, 73–98.

[HH] Hennion, Hubert ; Herve, Loıc . Limit theorems for Markov chains andstochastic properties of dynamical systems by quasi-compactness. LectureNotes in Mathematics, 1766. Springer-Verlag, Berlin, 2001.

[HIK] Hajian, Arshag; Ito, Yuji; Kakutani, Shizuo: Invariant measures and orbitsof dissipative transformations. Advances in Math. 9, 52–65. (1972).

[IT-M] Ionescu-Tulcea, C., Marinescu, G. Theorie ergodique pour des classesd’operations non completement continues Ann. Math. 47 (1950) 140-147.

[KS] Katsuda, A.; Sunada, T.: Closed orbits in homology classes. Inst. Hautes

Etudes Sci. Publ. Math. No. 71 (1990), 5–32.[LS] Ledrappier, F.; Sarig, O. Unique ergodicity for non-uniquely ergodic horo-

cycle flows. Discrete Contin. Dyn. Syst. 16 (2006), no. 2, 411–433.[LM] Lind, Douglas ; Marcus, Brian . An introduction to symbolic dynamics

and coding. Cambridge University Press, Cambridge, 1995.[L1] Livsic, A. N. On the spectra of adic transformations of Markov compact

sets. (Russian) Uspekhi Mat. Nauk 42 (1987), no. 3(255), 189–190; transl.Russian Math. Surveys 42 (1987), no. 3, 222223.

[L2] Livsic, A. N., Certain properties of the homology of Y -systems. Mat.Zametki 10 (1971), 555–564. Engl. Transl. in Math. Notes 10 (1971),758–763.

[Mrc] Marcus, B.: Unique ergodicity of the horocycle flow: variable curvaturecase. Israel J. Math. 21 (1975), 133–144.

[Mrg] Margulis, G. A.: Certain measures that are connected with U-flows oncompact manifolds. (Russian) Funkcional. Anal. i Prilozen. 4 1970 no. 1,62–76.

[N] Nagaev, S. V. Some limit theorems for stationary Markov chains. TheoryProbab. Appl. 2 (1957 ) 378-406.


[PP] Parry, William; Pollicott, Mark: Zeta functions and the periodic orbitstructure of hyperbolic dynamics. Asterisque No. 187–188 (1990), 268pp.

[PS] Parry, W. and Schmidt, K.: Natural coefficients and invariants forMarkov-shifts. Invent. Math. 76, 15-32 (1984)

[Po] Pollicott, M.: Zd-covers of horosphere foliations. Discrete Contin. Dynam.Systems 6 (2000), no. 1, 147–154,

[R] Ratner, M.: The central limit theorem for geodesic flows on n-dimensionalmanifolds of negative curvature. Israel J. Math. 16 (1973), 181–197.

[RE] Rousseau-Egele, J. Un theoreme de la limite locale pour une classe detransformations dilatantes et monotones par morceaux. Ann. Probab. 11(1983), no. 3, 772–788.

[Sa] Sarig, O.: Invariant Radon measures for horocycle flows on abelian covers.Invent. Math. 157 (2004), no. 3, 519551.

[S] Stone, Charles . On local and ratio limit theorems. 1967 Proc. Fifth Berke-ley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66),Vol. II: Contributions to Probability Theory, Part 2 pp. 217–224 Univ.California Press, Berkeley, Calif.

[TZ] Thaler M., Zweimuller R., Distributional limit theorems in infinite ergodictheory, Probab. Theory Relat. Fields 135, (2006), 15–52.

[V] Vershik, A. M. A new model of the ergodic transformations. Dynami-cal systems and ergodic theory (Warsaw, 1986), 381–384, Banach CenterPubl., 23, PWN, Warsaw, 1989.

(Jon. Aaronson) School of Math. Sciences, Tel Aviv University,69978 Tel Aviv, Israel.

E-mail address: [email protected]

(Omri Sarig) Faculty of Mathematics and Computer Sciences, TheWeizmann Institute for Science, POB 26, Rehovot 76100, Israel

E-mail address: [email protected]

EXPONENTIAL CHI SQUARED DISTRIBUTIONS IN INFINITE … · 2016. 7. 19. · EXPONENTIAL CHI SQUARED...

Documents

Transcript of EXPONENTIAL CHI SQUARED DISTRIBUTIONS IN INFINITE … · 2016. 7. 19. · EXPONENTIAL CHI SQUARED...