A Borel-Cantelli lemma and its applications January 30, 2012 2 1 0 NUNOLUZIA 2 Universidade Federal do Rio de Janeiro, Brazil n e-mail address: nuno@im.ufrj.br a J 7 ABSTRACT 2 ] WegiveaversionoftheBorel-Cantellilemma. Asanapplication,weproveanalmostsurelo- R calcentrallimittheorem. Asanotherapplication,weproveadynamicalBorel-Cantellilemma P for systems with sufficiently fast decay of correlations with respect to Lipschitz observables. . h t a Keywords: Borel-Cantellilemma;almostsurelocalcentrallimittheorem;decay m of correlations [ 1 v 6 Contents 6 8 5 1 Introduction and statements 1 . 1 1.1 A Borel-Cantelli Lemma . . . . . . . . . . . . . . . . . . . . . . . 2 0 1.2 An almost sure local central limit theorem . . . . . . . . . . . . . 2 2 1.3 A dynamical Borel-Cantelli Lemma . . . . . . . . . . . . . . . . . 4 1 : 1.4 Quantitative recurrence . . . . . . . . . . . . . . . . . . . . . . . 6 v i X 2 Proofs 7 2.1 Proofs of 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 r a 2.2 Proofs of 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Proofs of 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Proofs of 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1 Introduction and statements The classical Borel-Cantelli lemmas are a powerful tool in Probability Theory and Dynamical Systems. Let (Ω, ,P) be a probability space and (A ) a se- n F quence of measurable sets in . These lemmas say that (see [7] for proofs): F (BC1) If ∞n=1P(An)<∞ then P(x∈An i.o.)=0. P 1 (BC2) IfthesetsAnareindependentand ∞n=1P(An)=∞thenP(x∈An i.o.)= 1. P (BC3) If the sets An are pairwise independent and ∞n=1P(An)=∞ then n 1 P i=1 Ai 1 a.s. n P(A ) → Pi=1 i P Here1 istheindicatorfunctionofthesetA . Notethat(BC3)implies(BC2), Ai i but the proof of (BC3) is more elaborated. 1.1 A Borel-Cantelli Lemma Theorem 1. Let X be non-negative random variables and S = n X . If i n i=1 i supEX < , ES and there exists γ >1 such that i n ∞ →∞ P (ES )2 n var(S )=O (1) n (logES )(loglogES )γ (cid:18) n n (cid:19) then S n 1 a.s. ES → n We see that Theorem 1 implies (BC3), because when A are pairwise inde- i pendent sets, X =1 and ES then var(S )= n var(X ) ES . A i Ai n →∞ n i=1 i ≤ n slightmodificationofTheorem1alsogivesaversionoftheStrongLawofLarge P Numbers without assuming the random variables are pairwise independent. Corollary 1. Let X be identically distributed random variables with EX =µ, i i EX2 < and S = n X . If X M, for some constant M > 0, and i ∞ n i=1 i i ≥ − there exists γ >1 such that P n2 E(X X ) µ2 =O . i j − (logn)(loglogn)γ 1≤Xi 0 and b R (we assume h with this prok∈pZerty isimaximal). Let S = n X and a ∈R. By abuse of P n i=1 i ∈ notation, when we write S =a√n we mean S =[(a√n nb)/h]h+nb. n n P − Theorem 2. Let X be i.i.d. h-lattice valued random variables with EX = 0, i i EX2 = σ2 > 0 and EX 3 < , and S = n X . Let n be an increasing sequience of positive int|egie|rs an∞d a R.nThen i=1 i i ∈ P (a) If ∞i=1ni−1/2 <∞ then P(Sni =a√ni i.o.)=0. P (b) If there exist A>0 and γ >1 such that n n An1/2 and n A 1i2(logi)(loglogi) 3(logloglogi) 2γ i+1− i ≥ i i ≤ − − − for all i, then ni=11{Sni=aσ√ni} h e−a2/2 a.s. (2) P n n−1/2 → √2πσ i=1 i Let ∆a be the quotiePnt between the left and right hand sides of (2). Then, n for every N >0 there exists C >0 such that, for every ǫ>0, sup P(|∆an−1|>ǫ)≤Cǫ−2(loglogn)−1(logloglogn)−γ. a [ N,N] ∈− Remark 1. Concerning the divergent case ∞i=1ni−1/2 =∞ and a =0. In [3] theauthorsprovethatifthereexistsanintegerm>0suchthatn n +n i+mj i j foreveryi,jthenP(S =0 i.o.)=1. In[5]tPhesameauthorsclaimtha≥tifthere ni exist A > 0 such that n n > An1/2 for every i then P(S = 0 i.o.) = 1, i+1− i i ni however it seems their proof is not correct (see p. 184, Equation (21)). Considering n =i2 and the ‘change of variable’ k =i2 we get the following i almost sure local central limit theorem. Corollary 2. With the same hypotheses of Theorem 2, n lo1gn 1{Sk=aσ√k}√1k → √2hπσ e−a2/2 a.s. k=1 X When a=0 this was provedin [4] (see also[6]). For a R this was proved, ∈ independently, in [13]. The version for random variables with density follows. Theorem 3. Let X be i.i.d. random variables having density function whose i Fourier tranform (or some positive integer power of it) is integrable, EX =0, i EX2 = σ2 > 0 and EX 3 < , and S = n X . Let a R and n be an i | i| ∞ n i=1 i ∈ i increasing sequence of positive integers satisfying P n A(logi)(loglogi)α i+1 1+ for all i n ≥ i i 3 for some A>0 and α>2. Then n 1 1 1 e a2/2 a.s. (3) n {Sni=aσ√ni} → √2π − i=1 X For the special case a=0, (3) holds with n =i. i Let ∆a be the quotient between the left and right hand sides of (3). Then, n for every N >0 there exists C >0 such that, for every ǫ>0, sup P(∆a 1 >ǫ) Cǫ 2(logn) 1(loglogn)1 α. | n− | ≤ − − − a [ N,N] ∈− Inabovetheoremwecanconsidersequencesoftypen =[eA(logi)2(loglogi)α]. i In particular, considering the sequence n = 2i and the ‘change of variable’ i k =2i we get the following almost sure local central limit theorem. Corollary 3. With the same hypotheses of Theorem 3, n lo1gn 1{Sk=aσ√k}k1 → √12πe−a2/2 a.s. k=1 X Remark 2. Using the same techniques (and the Berry-Esseen Theorem) we can prove Theorem 3 with ‘ ’ instead of ‘=’ in (3), for i.i.d. random variables ≤ X with finite third moment, thus giving a new proofof the almost sure central i limit theorem (for related results, see [11] and references therein). 1.3 A dynamical Borel-Cantelli Lemma We want to consider the dynamical version of (BC3). Let (X,d) be a metric space, µ be a Borel probability measure on X, and T be a µ-preserving trans- formation on X. Let (B ) be a sequence of measurable sets in X. In what n conditions does (DBC) in=−011Bi(Tix) 1 for µ a.e. x P in=−01µ(Bi) → holds? We can easily find sufficiePnt conditions for (DBC) to hold: The sets B are all equal to B, µ(B)>0 and µ is ergodic. n • The sets T nB are pairwise independent and µ(B )= . − n n • ∞ The first one follows from Birkhoff ergodic theoremP, and the second one by (BC3)since 1 (Tix)=1 (x). However,itis veryunlikely fora dynamical Bi T−iBi system (T,X,µ) to have T nB pairwise independent. A far more reasonable − n condition is to have some sufficiently fast decay of correlations. In this section we extend some results of [10] where the same kind of problem is treated. For related results we refer the reader to [10] and references therein. We denote by the usual Lipschitz norm. We say that (T,X,µ,d) Lip k·k has polynomial decay of correlations (for Lipschitz observables) if, for every Lipschitz functions ϕ,ψ: X R, → ϕ Tnψdµ ϕdµ ψdµ c(n) ϕ ψ (4) Lip Lip ◦ − ≤ k k k k (cid:12)Z Z Z (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 4 where c(n) Cn α for some constants C >0 and α>0 (rate). − ≤ We say that (T,X,µ,d) has β-exponential decay of correlations, β > 0, if c(n) Ce αnβ, for some constants C,α>0, in(4). For β =1 we getthe usual − ≤ definition of exponential decay of correlations. For β < 1 this is also known as streched exponential decay of correlations. Wewillassumethefollowingconditiononthemeasureµ. ThereexistC >0, δ >0 (δ <2) and r >0 such that for all x X, 00). 0 0 7→ Example 1. 1. IfX isa compactmanifoldand µ isabsolutelycontinuouswithrespectto Lebesgue measure with density in L1+α for some α > 0, then µ satisfies (A). 2. If X = [0,1]2 and µ = ν µ where µ are absolutely continuous with x x × respect to Lebesgue measure with densities uniformly bounded in L , ∞ then there exists C > 0 such that the µ measure of any annulus of inner radius r and width ǫ is bounded by C√ǫ, and so µ satisfies (A). 3. If X = [0,1] and µ is the usual measure supported on the middle-third Cantor set, then F(x) = µ([0,x]) is the Devil’s staircase which is Hölder continuous with exponent log2/log3, and so µ satisfies (A). Theorem 4. Suppose µ satisfies (A) and Bi are balls such that in=−01µ(Bi)≥ nβ(logn)γ,forsome0<β <1andγ >1+(2−δ)β. If(T,X,µ,d)haspolynomial 2+δ P decay of correlations with rate α= 2/δ+1 1 then (DBC) is satisfyed. β − Let ∆ be the left hand side of (DBC). Then, for every ǫ > 0, ρ > 1 and n 0 < θ < γ(2+2βδ)−1 + 2δ −1, there exist C > 0 and Cθ > 0 (Cθ does not depend on ǫ,ρ) such that µ(∆n 1 >ǫ) C(logn)−1(loglogn)−ρ+Cθ(logn)−θ. | − | ≤ Corollary 4. Suppose µ satisfies (A) and B are balls such that µ(B ) i i ≥ i−β(logi)γ, for some 0 < β < 1 and γ > 1+ (2−δ2)+(1δ−β). If (T,X,µ,d) has polynomial decay of correlations with rate α= 2/δ+β then (DBC) is satisfyed. 1 β − Corollary 4 is stronger than Theorem 3.1 of [10]. Theorem 5. Suppose µ satisfies (A) and Bi are balls such that in=−01µ(Bi)≥ (logn)β(loglogn)(logloglogn)γ, for some β > 0, γ > 1. If (T,X,µ,d) has P β 1-exponential decay of correlations then (DBC) is satisfyed. − Let ∆ be the left hand side of (DBC). Then, for every ǫ > 0, there exists n C >0 such that µ(∆n 1 >ǫ) C(loglogn)−1(logloglogn)−γ. | − | ≤ 5 Corollary 5. Suppose µ satisfies (A) and B are balls such that µ(B ) i i ≥ i 1(logi)β 1(loglogi)(logloglogi)γ, for some β > 0, γ > 1. If (T,X,µ,d) − − has β 1-exponential decay of correlations then (DBC) is satisfyed. − Corollary 5 with β =1 is stronger than Theorem 4.1 of [10]. Many ‘nonuniformly hyperbolic dynamical systems’ exhibit some kind of decay of correlations, see [10] for examples with polynomial and exponential decay of correlations. The ‘Viana-like maps’ are a class of dynamical systems which possess streched exponential decay of correlations (but no exponential decay of correlations), see [1], so our results also give good recurrence results for these systems. 1.4 Quantitative recurrence As before, let (T,X,µ)be a measurepreservingtransformationof aprobability space X which is also endowed with a metric d. For α > 0, we denote by α H the α-Hausdorff measure of (X,d). One of the most beautiful results on the recurrence of dynamical systems is the following. Theorem 6. (Boshernitzan [2]) If, for some α>0, α is σ-finite on X, then H liminfnα1 d(Tn(x),x)< for µ a.e. x X. n ∞ ∈ →∞ If, moreover, α(X)=0, then H liminfnα1 d(Tn(x),x)=0 for µ a.e. x X. n ∈ →∞ Also a very nice result in this direction is as follows. Given x X, let 0 d¯ (x ) be the upper pointwise dimension of µ at x defined by ∈ µ 0 0 logµ(B(x ,r)) d¯ (x )=limsup 0 µ 0 logr r 0 → whereB(x ,r)istheballcenteredatx ofradiusr. Wealsosaythat(T,X,µ,d) 0 0 has superpolynomial decay of correlations (for Lipschitz observables) if it has polynomial decay of correlations with rate α for every α>0. Theorem 7. (Galatolo [9]) If (T,X,µ,d) has superpolynomial decay of correlations, then, for every x X and α>d¯ (x ), 0 µ 0 ∈ liminfnα1 d(Tn(x),x0)=0 for µ a.e. x X. n ∈ →∞ The dynamical Borel-Cantelli lemmas (for nested balls) stated in previous sectiongive,underadditionalassumptions,quantitativeversionsoftheseresults. Let Θα(x ) be the lower α-density of µ at x defined by µ 0 0 µ(B(x ,r)) Θα(x )=liminf 0 . µ 0 r 0 rα → So 0 Θα(x ) . ≤ µ 0 ≤∞ Theorem 8. 6 (a) If (T,X,µ,d) has polynomial decay of correlations with rate ϑ > 0, then, for every x X satisfying (A) with δ(x ) > 2/ϑ and µ( x ) = 0, 0 0 0 ∈ { } β = ϑ−2/δ(x0), γ >1+ (2−δ(x0))(1−β), there exists κ(n) with ϑ+1 2+δ(x0) logκ(n) limsup 0 (5) logn ≤ n →∞ such that, with α=d¯ (x ), µ 0 # 1 n N : nαβ d(Tn(x),x0)<κ(n) lim ≤ ≤ =θ(1 β) 1 (6) N→∞ n N1−β(logN)γ o − − (θ =1) for µ a.e. x X. ∈ If, moreover, Θα(x ) > 0, then (6) holds with κ(n) = (logi)γ/α and θ = µ 0 Θα(x ). µ 0 In particular, if (T,X,µ,d) has superpolynomial decay of correlations, then, for every x X satisfying (A) and µ( x ) = 0, β < 1, γ > 0 0 ∈ { } 1+ (2−δ(x0))(1−β), (6) holds. 2+δ(x0) (b) If (T,X,µ,d)has β-exponential decay of correlations, then, for every x 0 ∈ X satisfying (A) and µ( x )=0, γ >1, there exists κ(n) satisfying (5) 0 such that, with α=d¯ (x{),} µ 0 # 1 n N : nα1 d(Tn(x),x0)<κ(n) ≤ ≤ lim =θβ (7) N n (logN)β−1(loglogN)γ o →∞ (θ =1) for µ a.e. x X. ∈ If, moreover, Θα(x )>0, then (7) holds with µ 0 κ(n)=(logi)(β−1−1)/α(loglogi)γ/α and θ =Θα(x ). µ 0 2 Proofs 2.1 Proofs of 1.1 Proof of Theorem 1. Given σ >0, Chebyshev’s inequality implies P(S ES >σES ) var(S )/(σES )2 n n n n n | − | ≤ so P(S ES >σES ) Cσ 2(logES ) 1(loglogES ) γ (8) n n n − n − n − | − | ≤ for some constant C > 0. Note that (8) implies S /ES 1 in probability. n n → To get a.s. convergence we have to take subsequences. Let 0 < θ < γ 1 and n =inf n:ES ek/(logk)θ . Let U =S and note that the definit−ion and k { n ≥ } k nk EX M imply ek/(logk)θ EU σEU ) C˜σ 2k 1(logk)θ γ k k k − − − | − | ≤ 7 fBoorresol-mCeanctoenllsitlaenmtmCã (>BC01.)Simoplie∞ks=P1P(U(|Uk −EUEUk>| σ>EσUEUi.ok.))=<0∞. ,Sianncde σthies k k k arbitrary, it follows that U /EUP 1 a|.s.−To get|S /ES 1 a.s., pick an ω k k n n so that U (ω)/EU 1 and obser→ve that if n n0. In this particular case let n be a sequence of even positive − i integers and let E denote the event S = a√n whose precise meaning is i ni i S =2[a√n /2] (this eliminates probability zero of E ). ni i i (a) We have P(Ei)=(cid:18) ni+2[an√ini/2] /2(cid:19)2−ni ∼rπ2 ni−1/2e−a2/2 by Stirling’s formula, (cid:0)and the result fo(cid:1)llows from (BC1). (b) First of all, we notice the first condition on the sequence n implies i (√nj √ni)−1 max 1,4A−1 foreveryj >i. Thiswillbeimportantbecause − ≤ { } 8 when we do approximation to the lattice sometimes we will have to use a ± 2(√nj √ni)−1 instead of a. Let a˜= a +2max 1,4A−1 . − | | { } Also, the first conditionon the sequence n implies there exists A >0 such i 1 that n A i2 for every i, and so i 1 ≥ n A2(logn)1/2(loglogn)3/2(logloglogn)γ ≤ ni−1/2 ≤A−21logn (9) i=1 X for some constant A >0. 2 We want to apply Theorem 1 to the random variables X˜ = 1 . Let S˜ = i Ei n ni=iX˜i. WehaveESñ ∼(2/π)1/2e−a2/2 ni=1ni−1/2 →∞. Toverifycondition (1) in Theorem 1 we have n var(X˜ ) ES˜ and, for i k2 and, for the second line of inequalities, we also assume ≥ m 1, m>k. To estimate ≥ (P(E E ) P(E )P(E )) (10) i j i j ∩ − (cid:12)(cid:12)1≤Xiν n j n i Then we see that n 1/2 1 a2 1 a˜3 j 1+ and R=1+O + + . (cid:18)nj −ni(cid:19) ≤ νn (cid:18)√νn nj −ni √nj(cid:19) Since ∞i=1P(Ei)/√ni ≤C1 <∞, this implies (10) is less than some constant C times 2 P (ES˜ )2 (ES˜ )2 ES˜ ES˜ n +a2 n + n +a˜3 n. ν √ν ν √ν n n n n 9 Case 2: n ν n j n i ≤ Clearly (10) is less than 1/2 C ea2/2 P(E )P(E ) nj 3 i j n n i,j (cid:18) j − i(cid:19) X n C4 P(Ei) (nj ni)−1/2 (11) ≤ − i=1 j X X Given i, let N be the number of j’s satisfying n n ν n . Then n i j n i i+N ≤ ≤ ≤ ν n andN+i C ν i(logi)1/2. Thefirstconditiononthesequencen implies n i 5 n i ≤ n n C (j2 i2) for all i < j, so applying Cauchy-Schwarz inequality we j i 6 − ≥ − get 1/2 1/2 N N+i N+i (nj ni)−1/2 C7 (j+i)−1 (j i)−1 − ≤    −  j=i+1 j=i+1 j=i+1 X X X     which is less than C (loglogn)1/2(logn)1/2. Then (11) is less than 8 C (loglogn)1/2(logn)1/2ES˜ =O (ES˜ )2/√ν , 9 n n n (cid:16) (cid:17) where we have used (9). Applying Theorem 1 we get ni=11{Sni=a√ni} 2 e−a2/2 a.s. P ni=1ni−1/2 →rπ Lattice random walks. SPince EX 3 < , we have the following local central i | | ∞ limit theorem with rates (see [8]): P(S =aσ√n)= h e a2/2 1+O 1 . (12) n − √2πnσ √n (cid:18) (cid:18) (cid:19)(cid:19) Here, C ,C ,... denote appropriateabsolute constantsthat mightdepend (con- 1 2 tinuously)ona,σ,handonthedistributionofX (thisincludestheconstantsin i O(·)). Let Ei denote the event Sni =aσ√ni, X˜i =1Ei and Sñ = ni=iX˜i. By (12) we have ESñ ∼h(√2πσ)−1e−a2/2 ni=1ni−1/2 →∞, ni=1varP(X˜i)≤ESñ and, for i

A Borel-Cantelli lemma and its applications

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