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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: [email protected] 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<j≤n(cid:16) (cid:17) (cid:18) (cid:19) then S n µ a.s. n → 1.2 An almost sure local central limit theorem Let X be independent random variables such that each X assume the values i i +1 and 1 with probabilities 1/2 and 1/2. Then S = n X is the simple − n i=1 i randomwalkontheline. Itiswellknownthatthesequenceofrandomvariables P 1 does not obey the law of large numbers. More precisely (see [12]), {Si=0} n 1 limsup 1 =√2 a.s. n √nloglogn {Si=0} →∞ Xi=1 and there exists a constant 0<γ < such that 0 ∞ √loglogn n liminf 1 =γ a.s. n √n {Si=0} 0 →∞ i=1 X 2 It is then natural to ask if 1 obeys the law of large numbers for some {Sni=0} increasing sequence n of even positive integers. i More generally, we consider i.i.d. random variables X which are h-lattice i valued, i.e. P(X = kh+b) = 1, for some h > 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, 0<r r and 0<ǫ 1, 0 0 0 ∈ ≤ ≤ (A) µ( x:r <d(x,x )<r+ǫ )<Cǫδ. 0 { } Inwhatfollows,ifweonlyconsidernestedballsB =B(x ,r )(r 0)centered i 0 i i → at a given point x then we only require (A) holds for the point x , in other 0 0 words, r µ(B(x ,r)) is Hölder continuous (with exponent δ(x )>0). 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 <ek/logkθ+M. Replacing n by n in (8) we i k k ≤ ≤ get P(U 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 n<n →then k k k k+1 → ≤ U (ω) S (ω) U (ω) k n k+1 . EU ≤ ES ≤ EU k+1 n k To show that the terms at the left and right end converge to 1, we rewrite the last inequalities as EU U (ω) S (ω) U (ω)EU k k n k+1 k+1 . EU EU ≤ ES ≤ EU EU k+1 k n k+1 k From this we see it is enough to show EU /EU 1. Since k+1 k → ek/(logk)θ EU EU e(k+1)/(log(k+1))θ +M k k+1 ≤ ≤ ≤ we must show e(k+1)/(log(k+1))θ/ek/(logk)θ converges to 1. Wenotethatek/(logk)θ =kh(k)whereh: (1, ) (0, ), h(x)=x/(logx)1+θ. ∞ → ∞ Then we must show (k+1)h(k+1) 1 h(k) = 1+ e(h(k+1) h(k))log(k+1) − kh(k) k (cid:18) (cid:19) converges to 1. Clearly h(k)=o(k) and so the left hand side of product above converges to e0 = 1. We note that 0 < h(x) < (logx) 1 θ and h (x) < 0 (for ′ − − ′′ allsufficientlylargex),sotherighthandsideofproductaboveis exp(log(k+ ≤ 1)/(logk)1+θ) which also converges to 1. 2.2 Proofs of 1.2 Proof of Theorem 2. The simple random walk. We treat this particular case separately because its proof is elementary. Then we say how it extends easily to lattice random walks by using a local central limit theorem. We will use Stirling’s formula n n 1 n!=√2πn 1+O e n (cid:16) (cid:17) (cid:18) (cid:18) (cid:19)(cid:19) where O(n 1)>0. 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<j, P i=1 i ≤P n P(Ei∩Ej)=P(SniP=a√ni)P(Snj−ni =2[a√nj/2]−2[a√ni/2]) 1/2 n =P(Sni =a√ni)P(Snj =a√nj) n jn R (cid:18) j − i(cid:19) 1/2 n =P(E )P(E ) j R i j n n (cid:18) j − i(cid:19) where, for n 2a˜6, j ≥ a2√ni 1 a˜3 R=e√nj+√ni 1+O + (cid:18) (cid:18)nj −ni √nj(cid:19)(cid:19) (for a=0 this holds with a˜=0) is obtained using Stirling’s formula and (1+k/m)m ek (1+k/m)m(1+k2/m), ≤ ≤ (1+k/m)m (1 k/m) m (1+k/m)m(1 k2/m) 1, − − ≤ − ≤ − where k 0, m > 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≤Xi<j≤n (cid:12)(cid:12) (cid:12) (cid:12) we separate the sum in two cases. Let √ν = (logES˜ )(loglogES˜ )γ. In all n n n cases we restrict to n 2a˜6 and ν a4. Here C ,C ,... denote appropriate i n 1 2 ≥ ≥ absolute constants (which do not depend on a). Case 1: n >ν 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<j, P P P(Ei∩Ej)=P(Sni =aσ√ni)P(Snj−ni =aσ(√nj −√ni)+O(1)) 1/2 n =P(Sni =a√ni)P(Snj =a√nj) n jn R (cid:18) j − i(cid:19) n 1/2 =P(E )P(E ) j R i j n n (cid:18) j − i(cid:19) where, for n C , j 1 ≥ a2√ni 1 R=e√nj+√ni 1+O . √n n (cid:18) (cid:18) j − i(cid:19)(cid:19) 10