EXPONENTIAL CHI SQUARED DISTRIBUTIONS IN INFINITE ERGODIC THEORY JON. AARONSON & OMRI SARIG Abstract. We prove distributional limit theorems for random walk adic transformations obtaining ergodic distributional limits of exponential chi squared form. 0 Introduction As in [A1], for X, ,m a σ-finite measure space, F n X 0, measurable, and Y 0, a random variable, we say that F n con- verges strongly in distribution to Y , (written F n d n Y,) if it converges in law with respect to all m-absolutely continuous probabilities; equiv- alently gF n n E gY weak in L m for each bounded, continuous function f 0, R. For discussion of strong distributional convergence, see [A1], [A2], [E] and [TZ]. Here, we study distributional stability. As in [A2], we’ll call a conservative, ergodic measure preserving transformations X, ,m,T distributionally stable if there are constants an 0 and a random variable Y on 0, so that 1 an S n f d Ymf f L 1 where S n f n 1 k 0 f T k and mf X fdm. By the ratio ergodic theorem, if the above convergence holds for some f L 1 , then it holds f L 1 . 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, grant 239885. 1
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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
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.
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.
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;
Exponential χ2 distributions 5
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]).
Exponential χ2 distributions 7
§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
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
Exponential χ2 distributions 9
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
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.
Exponential χ2 distributions 11
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
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
Exponential χ2 distributions 19
§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,
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]:
Exponential χ2 distributions 21
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.
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(Jon. Aaronson) School of Math. Sciences, Tel Aviv University,69978 Tel Aviv, Israel.