Limit theorems for numbers of returns in arrays under ϕ-mixing

arXiv:1910.07908v1 [math.DS] 16 Oct 2019 LIMIT THEOREMS FOR NUMBERS OF RETURNS IN ARRAYS UNDER φ-MIXING YURI KIFER INSTITUTE OF MATHEMATICS HEBREW UNIVERSITY JERUSALEM, ISRAEL Abstract. We consider a φ-mixing shift T on a sequence space Ω and study a the number NN of returns {T qN (n) ω ∈ Aa n } at times qN (n) to a cylinder An constructed by a sequence a ∈ Ω where n runs either until a fixed integer N or until a time τN of the first return {T qN (n) ω ∈ Abm } to another cylinder Abm constructed by b ∈ Ω. Here qN (n) are certain functions of n taking on nonnegative integer values when n runs from 0 to N and the dependence on N is the main generalization here in comparison to [16]. Still, the dependence on N requires certain conditions under which we obtain Poisson distributions limits of NN when counting is until N as N → ∞ and geometric distributions limits when counting is until τN as N → ∞. The results and the setup are similar to [13] where multiple returns are considered but under the stronger ψ-mixing assumption. 1. Introduction The study of returns to (hits of) shrinking targets by a dynamical system, started in [17], [9] and [6], has already about 30 years history. These works were extended in various directions, in particular, to returns to shrinking geometric balls by uniformly and non-uniformly hyperbolic dynamical systems (see, for instance, [11] and references there), to multiple returns to shrinking cylinder sets under ψ-mixing (see, for instance, [14]) and others. More recently, motivated by the research on open dynamical systems (see, for instance, [5]) the asymptotic behaviour of numbers of returns to a shrinking target until the first arrival to another shrinking target was investigated in [15] and [16] where the first work dealt with the ψ-mixing case while the second one dealt with a φ-mixing situation which allowed applications to a wider class of dynamical systems. Another generalization started in Ch.3 of [10] and continued in [13] and [7], dealt with returns at prescribed times which depended also on the total observation time where additional peculiarities appeared. In this paper we consider two related types of limit theorems for numbers of returns which are represented by the sums SN = N X IAan ◦ T qN (k) N and ΣN = k=1 τN X IAan ◦ T qN (k) N k=1 Date: December 13, 2021. 2000 Mathematics Subject Classification. Primary: 60F05 Secondary: 37D35, 60J05. Key words and phrases. Geometric distribution, Poisson distribution, numbers of returns, φmixing, shifts. 1 2 Yu.Kifer where IΓ is the indicator of a set Γ, Aan is a cylinder set of the length n constructed by a sequence a, T is a φ-mixing left shift on a sequence space Ω, τN (ω) is the first k such that T qN (k) ω belongs to another cylinder set AbmN and qN (k) are certain functions taking on nonnegative integer values when k runs from 1 to N for N = 1, 2, .... In probability such sums where summands themselves depend on the number of summands are usually called (triangular) arrays. We will provide conditions on functions qN (k) such that as N → ∞ the sum SN converges in distribution to a Poisson random variable while the sum ΣN converges in distribution to a geometric random variable. It is easy to see that without certain conditions such results do not hold, in general. Indeed, taking qN (k) = k(N − k) we obtain that the above sums may converge only to a random variable taking on just even values, and so the limits cannot have Poisson or geometric distributions. Our results remain valid for dynamical systems possessing appropriate symbolic representations such as Axiom A diffeomorphisms (see [3]), expanding transformations and some maps having symbolic representations with an infinite alphabet and a ψ-mixing invariant measure such as the Gauss map with its Gauss invariant measure and more general f -expansions (see [8]). A direct application of the above results in the symbolic setup yields the corresponding results for arrivals to elements of Markov partitions but employing additional technique (see, for instance, [16]) it is not difficult to extend these results for arrivals to shrinking geometric balls. Since we assume only φ-mixing, rather than ψ-mixing, our results remain valid for some classes of nonuniformly expanding maps of the interval such as Gibbs-Markov maps and some others (cf. [16]). In the probability direction we can consider Markov chains with countable state spaces satisfying the Doeblin condition which are known to be exponentially fast φ-mixing (see [4]), and so our results are applicable to the corresponding shifts in the path spaces. 2. Preliminaries and main results Our setup consists of a finite or countable set A which is not a singleton, the sequence space Ω = AN , the σ-algebra F on Ω generated by cylinder sets, the left shift T : Ω → Ω, and a T -invariant probability measure P on (Ω, F ) which is assumed to be φ-mixing with respect to the σ-algebras Fmn , n ≥ m generated by the cylinder sets of the form {ω = (ω0 , ω1 , ...) ∈ Ω : ωi = ai for m ≤ i ≤ n} for some am , am+1 , ..., an ∈ A. Observe also that Fmn = T −m F0,n−m for n ≥ m. Recall, that the φ-dependence (mixing) coefficient between two σ-algebras G and H can be written in the form (see [4]),  (2.1) φ(G, H) = supΓ∈G,∆∈H P P(Γ∩∆) (Γ) − P (∆) , P (Γ) 6= 0 = 1 2 sup{kE(g|G) − E(g)kL∞ : g is H − measurable and kgkL∞ ≤ 1}. Set also φ(n) = sup φ(F0,m , Fm+n,∞ ). m≥0 The probability P is called φ-mixing if φ(n) → 0 as n → ∞. We will need also the α-dependence (mixing) coefficient between two σ-algebras G and H which can be written in the form (see [4]),  α(G, H) = supΓ∈G,∆∈H P (Γ ∩ ∆) − P (Γ)P (∆) (2.2) = 1 4 sup{kE(g|G) − E(g)kL1 : g is H − measurable and kgkL∞ ≤ 1}. Numbers of returns 3 Set also α(n) = sup α(F0,m , Fm+n,∞ ). m≥0 For each word a = (a0 , a1 , ..., an−1 ) ∈ An we will use the notation [a] = {ω = (ω0 , ω1 , ...) : ωi = ai , i = 0, 1, ..., n − 1} for the corresponding cylinder set. Without loss of generality we assume that the probability of each 1-cylinder set is positive, i.e. P ([a]) > 0 for every a ∈ A, and since A is not a singleton we have also supa∈A P ([a]) < 1. Write ΩP for the support of P , i.e. ΩP = {ω ∈ Ω : P [ω0 , ..., ωn ] > 0 for all n ≥ 0}. For n ≥ 1 set Cn = {[w] : w ∈ An }. Then F0,n consists of ∅ and all unions of disjoint elements from Cn+1 . If the probability P is φ-mixing then by Lemma 3.1 from [16] there exists υ > 0 such that (2.3) P (A) ≤ e−υn for all n ≥ 1 and A ∈ Cn . Next, for any U ∈ F0,n−1 , U 6= ∅ define π(U ) = min{k ≥ 1 : U ∩ T −k U 6= ∅} and observe that π(U ) ≤ n. We will be counting the returns to U at times qN (k) considering the sum N X U SN = IU ◦ T qN (k) . k=1 Our goal will be to show that if U is replaced by a sequence of Borel sets UN ⊂ Ω UN such that N P (UN ) converges as N → ∞ then SN converges in distribution to a Poisson random variable and, as an example in Introduction shows, in order to achieve this some assumptions on functions qN (k) are necessary. 2.1. Assumption. qN (n) is a function taking on nonnegative integer values on integers n, N ≥ 0, defined arbitrarily when n > N and such that for some constant K > 0 and all N ≥ 1 the following properties hold true: (i) For all k the number of integers n, 0 ≤ n ≤ N satisfying the equation qN (n) = k does not exceed K; (ii) The number of pairs m 6= n satisfying 0 ≤ m, n ≤ N and solving the equation qN (n) − qN (m) = 0 does not exceed K; First, note that the example of qN (k) = k(N − k) from Introduction does not satisfy Assumption 2.1(ii) since qN (n) = qN (N − n), and so at least [N/2] − 1 pairs n 6= m = N − n solve the equation in (ii). Next, observe that if there exist n0 , N0 ≥ 1 such that for all N ≥ N0 the function qN (n) of n is strictly increasing when n0 ≤ n ≤ N , then the whole Assumption 2.1 is satisfied. Indeed, at most one n ≥ n0 can solve the equation qN (n) = k when N and k are fixed, and so the number of solutions in (i) cannot exceed n0 + 1. Next, if N ≥ n, m ≥ n0 then qN (n) = qN (m) will hold true only if n = m. If, say, m < n0 then for such m there could be at most one n ≥ n0 satisfying qN (n) = qN (m). It follows that there exist at most n20 + 2n0 pairs 0 ≤ m, n ≤ N such that qN (n) = qN (m). In particular, if qN (n) = r(n) + g(N ) where r is a nonconstant polynomial in n and g is a function 4 Yu.Kifer of N , both nonnegative for n, N ≥ 0 and taking on integer values on integers, then qN satisfies Assumption 2.1. Indeed, the number of solutions in Assumption 2.1(i) is bounded by the degree of r and there exists an integer n0 ≥ 1 such that the polynomial r is strictly increasing on [n0 , ∞). For any two random variables or random vectors Y and Z of the same dimension denote by L(Y ) and L(Z) their distribution and by dT V (L(Y ), L(Z)) = sup |L(Y )(G) − L(Z)(G)| G the total variation distance between L(Y ) and L(Z) where the supremum is taken over all Borel sets. Denote by Pois(λ) the Poisson distribution with a parameter k λ > 0, i.e. Pois(λ)(k) = e−λ λk! for each k = 0, 1, 2, .... Our first result is the following. 2.2. Theorem. Suppose that Assumption 2.1 is satisfied. Then there exists a constant C ≥ 1 such that for any n, V ∈ F0,n−1 , N and R, (2.4) V dT V (L(SN ), Pois(λN )) ≤ CN R(P (V ))2
PR +P (V ) r=π(V ) (φ([r/2] + 1) + P (T n−[r/2] V )) + φ(R − n) where λN = N (P (V ))ℓ . We observe that related estimates under φ-mixing were obtained in [1] but it is difficult to obtain definitive convergence results from there. 2.3. Corollary. Suppose that Assumption 2.1 is satisfied and the φ-mixing coefficient is summable, i.e. ∞ X φ(k) < ∞. k=1 Let VL ∈ F0,nL −1 , L = 1, 2, ... be a sequence of sets such that nL P (VL ) → 0 and PnL −1 nL −r VL ) → 0 as L → ∞. Let NL → ∞ as L → ∞ be a sequence of r=π(VL ) P (T integers such that 0 < C −1 ≤ λL = NL P (VL ) ≤ C < ∞ for some constant C and all L ≥ 1. Then (2.5) VL dT V (L(SN ), Pois(λL )) → 0 as L → ∞ L VN and if limL→∞ λL = λ then the distribution of SNLL converges in total variation as L → ∞ to the Poisson distribution with the parameter λ. In particular, if VL = AηnL = [η0 , ..., ηnL −1 ] = {ω ∈ Ω : ω0 = η0 , ..., ωnL −1 = ηnL −1 } with nL → ∞ as L → ∞ and η ∈ ΩP is nonperiodic then π(AηnL ) → ∞ as L → ∞ and the above statements hold true for such VL ’s provided the above conditions on λL are satisfied. Next, for any V ∈ F0,n−1 , V 6= ∅ and W ∈ F0,m−1 , W 6= ∅ define π(V, W ) = min{k ≥ 1 : V ∩ T −k W 6= ∅ or W ∩ T −k V 6= ∅} . It is clear that π(V, W ) ≤ m ∧ n, and so κV,W = min{π(V, W ), π(V ), π(W )} ≤ m ∧ n where, as usual, for n, m ≥ 1 we denote m∨n = max{m, n} and m∧n = min{m, n}. Set τW (ω) = min{k ≥ 1 : T qN (k) ω ∈ W } Numbers of returns 5 with τW (ω) = ∞ if the event in braces does not occur and define ΣV,W N = τW X IV ◦ T qN (k) . k=1 Denote by Geo(ρ) the geometric distribution with a parameter ρ ∈ (0, 1), i.e. Geo(ρ)(k) = ρ(1 − ρ)k for each k = 0, 1, 2, .... 2.4. Theorem. Assume that Assumption 2.1 is satisfied. Then there exists a constant C > 0 such that for any disjoint sets V ∈ F0,n−1 and W ∈ F0,m−1 with P (V ), P (W ) > 0 and all integers n, m, N, R ≥ 1, 
(2.6) dT V (L ΣV,W ), Geo(ρ) ≤ C (1 − P (W ))N + (n ∨ m)(P (V ) + P (W )) N +RN (P (V ) + P (W ))2 + N φ(R − n ∨ m) 
Pn∨m−1 n∨m−r n∨m−r V ) + (P (T W) +N (P (V ) + P (W )) r=κV,W (φ(r) + P (T where ρ = P (W ) P (V )+P (W ) . 2.5. Corollary. Suppose that Assumption 2.1 holds true and the φ-mixing coefficient is summable. Let VL ∈ F0,nL −1 and WL ∈ F0,mL −1 , L = 1, 2, ... be two sequences of sets such that (2.7) (2.8) (nL ∨ mL )(P (VL ) + P (WL )) → 0, κVL ,WL → ∞ αL = nL ∨m XL −1 as L → ∞, (P (T nL ∨mL −r VL ) + P (T nL ∨mL −r WL )) → 0 as L → ∞ r=κVL ,WL and for some constant C and all L ≥ 1, 0 < C −1 ≤ (2.9) P (VL ) ≤ C < ∞. P (WL ) Let NL , L = 1, 2, ... be a sequence satisfying (2.10) NL P (WL ) → ∞ and NL (MNL + nL ∨ mL + αL )(P (WL ))2 → 0 as L → ∞ (ε) where MN = MN = min{n ≥ 1 : nφ(n). Then (2.11) n γ ε (n) ≥ N } for some 0 < ε < 1 and γ(n) = dT V (L(ΣVNLL,WL ), Geo(ρL )) → 0 as L → ∞ where ρL = P (WL )(P (WL ) + P (VL ))−1 . In particular, if limL→∞ ρL = ρ, then ΣVNLL,WL converges in total variation as L → ∞ to the geometric distribution with the parameter ρ. Furthermore, let VL = AξnL = [ξ0 , ..., ξnL −1 ] ∈ CnL and WL = AηmL = [η0 , ..., ηmL −1 ] ∈ CmL with nL , mL → ∞ as L → ∞ and suppose that ξ, η are not periodic and not shifts of each other. Then (2.12) κAξn L ,Aη mL nL ∧ mL + κAξn L ,Aη mL → ∞ as L → ∞ and if also (2.13) − nL ∨ mL → ∞ as L → ∞ 6 Yu.Kifer then (2.8) holds true. In fact, (2.13) is satisfied for P × P -almost all (ξ, η) ∈ Ω × Ω provided 2nL ∧ mL − nL ∨ mL − 3υ ln(nL ∧ mL ) → ∞ as L → ∞ (2.14) where υ is from (2.3). Observe that when qN (n) does not depend on N then ΣV,W does not depend N on N either and in order to obtain (2.11) relying on (2.6) we have only to pick up some sequence NL satisfying (2.10) which is always possible provided (2.7)–(2.9) hold true. 3. Poisson distribution limits We will need the following semi-metrics between positive integers k, l > 0, δN (k, l) = |qN (k) − qN (l)|. It follows from Assumption 2.1(i) that for any integers n ∈ {1, ..., N } and k ≥ 0, #{m : δN (n, m) = k} ≤ 2K. (3.1) For any integers M, R ≥ 1 and 1 ≤ n ≤ N introduce the sets M,R N,R R Bn,N = {l : 1 ≤ l ≤ M, δN (l, n) < R} and Bn,N = Bn,N . By (3.1), for any n, M,R ≤ min(M, 2KR). #Bn,N (3.2) V V Let V ∈ F0,n−1 and set Xk,N = Xk,N = IV ◦ T qN (k) . Then SN = SN = PN X . Set p = P {X = 1} and p = P {X = 1 and X k,N k,N k,N k,l,N k,N l,N = k=1 1}. Since T is P -preserving pk,N = E(IV ◦ T qN (k) ) = P (V ) and pk,l,N = P (V ∩ T −(qN (k)−qN (l)) V ) provided qN (l) ≤ qN (k). By Theorem 1 from [2] we obtain dT V (L(SN ), P ois(λN )) ≤ b1 + b2 + b3 (3.3) where b1 , b2 and b3 are defined by (3.4) b1 = N X X pn,N pl,N , b2 = N X X pn,l,N n=1 n6=l∈B R n=1 l∈B R n,N n,N and (3.5) b3 = N X R sn,N with sn,N = E|E(Xn,N − pn,N |σ{Xl,N : l ∈ {1, ..., N } \ Bn,N })|. n=1 By (3.2) and (3.4) we conclude that (3.6) b1 = N X X k=1 pk,N pl,N ≤ 2KRN (P (V ))2 . R l∈Bk,N In order to estimate pk,l,N we observe that if |i − j| < π(V ) then (IV ◦ T i )(IV ◦ T ) = 0. Hence, pk,l,N = 0 if δN (k, l) < π(V ). Now suppose that δN (k, l) = d with π(V ) ≤ d < n. Then j (3.7) either qN (l) ≤ qN (k) − d or qN (l) ≥ qN (k) + d. Numbers of returns 7 Assume, for instance, that the first inequality in (3.7) holds true and let r = qN (k) − qN (l). Then r ≥ d ≥ π(V ). If r ≥ n then since V ∈ F0,n−1 , we obtain by the definition of the φ-mixing coefficient that pk,l,N = P (V ∩ T −r V ) ≤ (φ(r − n + 1) + P (V ))P (V ). (3.8) Suppose that π(V ) ≤ r < n and assume that V ∩ T −r V 6= ∅. Let s ≥ n − r, set Vs = T s V and observe that T −s Vs ⊃ V . Then by the definition of the φ-mixing coefficient, (3.9) pk,l,N = P (V ∩ T −r V ) ≤ P (V ∩ T −(r+s) Vs ) ≤ (φ(r + s − n + 1) +P (T s V ))P (V ) ≤ (φ([r/2] + 1) + P (T n−[r/2] V ))P (V ) taking s = n − [r/2]. If the second inequality in (3.7) holds true then we obtain (3.8) if r = qN (l) − qN (k) ≥ n, while if π(V ) ≤ r < n then we arrive at (3.9). Observe that by Assumption 2.1(i) for any N ≥ 1 and integers k ≥ 0 and r, #{l ≥ 0 : qN (k) − qN (l) = r} ≤ K. (3.10) Now, it follows from (3.1), (3.2) and (3.8)–(3.10) that (3.11) b2 = N X X pk,l,N ≤ 4KN P (V ) R k=1 k6=l∈Bk,N R X (φ([r/2] + 1) + P (T n−[r/2] V )). r=π(V ) R Next, we estimate sk,N and b3 defined by (3.5). Since δN (k, l) ≥ R for l 6∈ Bk,N and V ∈ F0,n−1 , we derive from Lemma 3.3 in [16] and the definition of the αmixing coefficient that for n < R < N ,
(3.12) sk,N ≤ α FqN (k),qN (k)+n , σ(F0,qN (k)−R+n , FqN (k)+R−n,∞ ) ≤ 3φ(R − n). Hence, by (3.5) and (3.12), (3.13) b3 = N X sk,N ≤ 3N φ(R − n). k=1 Finally, collecting (3.3), (3.6), (3.11) and (3.13) we derive (2.4) completing the proof of Theorem 2.2.  Next, we will derive Corollary 2.3 from the estimate (2.4). The first part of Corollary 2.3 would follow if we find an integer valued sequence RL , L = 1, 2, ... RL → 0 and NL φ(RL − nL ) → 0 as L → ∞. In order to do such that RL → ∞, N L this we observe first that since φ(k) is summable and nonincreasing, [N/2]φ(N ) ≤ N X φ(k) ≤ k=[N/2] X φ(k) → 0 as N →∞ k=[N/2] which means that γ(N ) = N φ(N ) → 0 as N → ∞. Observe that if φ(k) = 0 for some k ≥ 1 then by monotonicity φ(n) = 0 for all n ≥ k. In this case there is nothing to prove taking, say, RL = 2nL → ∞ as L → ∞. Hence, we can and will assume that φ(n) > 0 for all n ≥ 1. For some 0 < ε < 1 set n (ε) MN = MN = min{n ≥ 1 : ε ≥ N } → ∞ as N → ∞. γ (n) Then MN ≥N γ ε (MN ) and N φ(MN ) = N γ(MN ) ≤ γ 1−ε (MN ) → 0 as N → ∞. MN 8 Yu.Kifer MN N 1 N −1 Since γ εM (MN −1) < N then MN −1 > γ ε (MN −1) → ∞ as N → ∞, and so N → 0 as N → ∞. Hence, taking RL = MNL + nL we conclude the proof of the first part of Corollary 2.3. In the second part of Corollary 2.3 we set VL = AηnL = [η0 , ..., ηnL −1 ] where η is a nonperiodic sequence and observe that P (AηnL ) ≤ e−υnL by (2.3). Hence, the conditions of the first part of Corollary 2.3 would hold true provided π(Aηn ) → ∞ as n → ∞ (3.14) whenever η is a nonperiodic sequence. To see this note that π(Aηn ) is, clearly, nondecreasing in n, and so limn→∞ π(Aηn ) = r exists. If r < ∞ then there exists n0 ≥ 1 such that π(Aηn ) = r for all n ≥ n0 which means that η is periodic with the period r. Hence, r = ∞ since η is not periodic completing the proof of Corollary 2.3.  4. Geometric distribution limits It will be convenient to set V (0) = V ∈ F0,n−1 , V (1) = W ∈ F0,m−1 and (α) Xk,N = IV (α) ◦ T qN (k) , α = 0, 1 so that (1) τ = τV (1) = min{k ≥ 1 : Xk,N = 1} and ΣVN (0) ,V (1) = τ X (0) Xk,N . k=1 Set also SL = Let (α) {Yk,N (0) PL k=1 Xk,N , so that Sτ = ΣVN (0) ,V (1) , and denote τN = min(τ, N ). : k ≥ 1, α = 0, 1} be a sequence of independent Bernoulli random (α) (α) variables such that Yk,N has the same distribution as Xk,N . Since P is T -invariant (α) (α) (α) (α) E(Xk,N ) = P {Xk,N = 1} = E(Yk,N ) = P {Yk,N = 1} = P (V (α) ). Set SL∗ = L X (0) (1) ∗ Yk,N , τ ∗ = min{k ≥ 1 : Yk,N = 1} and τN = min(τ ∗ , N ). k=1 We can and will assume that all above random variables are defined on the same (sufficiently large) probability space. By Lemma 3.1 from [14] the sum Sτ∗∗ has the geometric distribution with the parameter P (V (1) ) >ρ P (V + P (V (0) )(1 − P (V (1) ))
−1 where ρ = P (V (1) ) P (V (1) ) + P (V (0) ) . Next, we can write (4.1) ̺= (1) ) dT V (L(Sτ ), Geo(ρ)) ≤ A1 + A2 + A3 + A4 (4.2) where A1 = dT V (L(Sτ ), L(SτN )), A2 = dT V (L(SτN ), L(Sτ∗∗ )), A3 N dT V (L(Sτ∗∗ ), L(Sτ∗∗ )) and A4 = dT V (Geo(̺), Geo(ρ)). N (α) Introduce random vectors XN (0) {XN , (1) XN }, (α) YN (α) {Yn,N , = (α) = {Xk,n , 1 ≤ k ≤ N }, α = 0, 1, XN = (0) (1) = 1 ≤ k ≤ N }, α = 0, 1 and YN = {YN , YN }. Observe that the event {Sτ 6= SτN } can occur only if τ > N . Also, we can write Numbers of returns 9 (1) (1) {τ > N } = {Xn,N = 0 for all k = 1, ..., N } and {τ ∗ > N } = {Yn,0 = 0 for all k = 1, ..., N } Hence, (4.3) (1) A1 ≤ P {τ > N } = P {τ ∗ > N } + |P {Xn,N = 0 for n = 1, ..., N } (1) −P {Yn,N = 0 for n = 0, 1, ..., N }| ≤ P {τ ∗ > N } + dT V (L(XN ), L(YN )). (1) Since Yk,N , k = 0, 1, ... are i.i.d. random variables we obtain that P {τ ∗ > N } = (1 − P (V (1) ))N . (4.4) Also A3 ≤ P {τ ∗ > N } = (1 − P (V (1) ))N . (4.5) The estimate of A4 is also easy P∞ P∞ (4.6) A4 ≤ k=0 |̺(1 − ̺)k − ρ(1 − ρ)k | ≤ 2 k=1 ((1 − ρ)k − (1 − ̺)k ) = 2(1 − ρ)ρ−1 − 2(1 − ̺)̺−1 = 2(̺−ρ) ρ̺ = 2P (V (1) ). Next, we observe that by Theorem 3 in [2], (4.7) A2 ≤ dT V (L(XN ), L(YN )) ≤ 2b1 + 2b2 + b3 + 2 (α) X (pk,N )2 1≤k≤N, α=0,1 (α) (α) where pk,N = P {Xk,N = 1} = P (V (α) ) while the definitions of b1 , b2 and b3 are similar to Section 3 taking into account the additional parameter α. Namely, setting (α) (β) R Bk,N = {(l, 0), (l, 1) : 1 ≤ l ≤ N, δ(k, l) < R}, pα,β k,l,N = E(Xk,N Xl,N ) and IN = {(k, α) : 1 ≤ k ≤ N, α = 0, 1} we have X X (α) (β) pk,N pl,N , (4.8) b1 = R (k,α)∈IN (l,β)∈Bk,N (4.9) b2 = X X (α,β) pk,l,N and R (k,α)∈IN (k,α)6=(l,β)∈Bk,N (4.10) b3 = X (α) sk,N where (k,α)∈IN (α)
(α) (α) (α) (β) R sk,N = E E Xk,N − pk,N |σ{Xl,N : (l, β) ∈ IN \ Bk,N } . Since pk,N = P (V (α) ), it follows taking into account (3.1) and (3.2) that (4.11) b1 ≤ 6KRN ((P (V (0) ))2 + (P (V (1) ))2 ). In order to estimate pα,β k,l,N (and, eventually, b2 ) we observe that (IV (0) ◦ T i )(IV (1) ◦ T j ) = 0 if |i − j| < κV (0) ,V (1) . Hence, pα,β k,l,N = 0 if δN (k, l) < κV (0) ,V (1) . Now suppose that δN (k, l) = d ≥ κV (0) ,V (1) . Then we have to deal with two alternatives from (3.7). Assume, for instance, that the first inequality in (3.7) holds true and let r = qN (k) − qN (l). Then r ≥ d ≥ κV (0) ,V (1) . If r ≥ n then since V (0) ∈ F0,n−1 and V (1) ∈ F0,m−1 , we obtain by the definition of the φ-mixing coefficient that (4.12) (β) pα,β ∩ T −r V (α) ) ≤ (φ(r − m + 1) + P (V (α) ))P (V (β) ). k,l,N = P (V 10 Yu.Kifer Suppose that κV (0) ,V (1) ≤ r < n and assume that V (β) ∩T −r V (α) 6= ∅. Let s ≥ n−r, set Vs = T s V and observe that T −s Vs ⊃ V . Then by the definition of the φ-mixing coefficient, (4.13) (α) (β) pα,β ∩ T −r V (α) ) ≤ P (V (β) ∩ T −(r+s) Vs k,l,N = P (V s ≤ (φ(r + s − n + 1) + P (T V ≤ (φ([r/2] + 1) + P (T n−[r/2] (α) V ))P (V (α) ))P (V (β) ) ) (β) ) taking s = n − [r/2]. If the second inequality in (3.7) holds true then we obtain (4.12) if r = qN (l) − qN (k) ≥ n, while if κV (0) ,V (1) ≤ r < n then we arrive at (4.13). and integers k ≥ 0 and r, Now, it follows from (3.1), (3.2), (3.10), (4.12) and (4.13) that P P α,β (0) (4.14) b2 = N ) + P (V (1) )) R , α,β=0,1 pk,l,N ≤ 4KN (P (V k=1 k6=l∈Bk,N PR × r=κ (0) (1) (φ([r/2] + 1) + P (T n−[r/2]V (0) ) + P (T m−[r/2] V (1) )). V ,V Similarly to (3.13) we obtain also that (4.15) b3 = N X (α) sk,N ≤ 6N φ(R − n ∨ m). 1≤k≤N, α=0,1 These provide the estimate of A2 by (4.7), (4.8)–(4.11), (4.14) and (4.15). Finally, combining (4.2)–(4.11), (4.14) and (4.15) we derive (2.6) completing the proof of Theorem 2.4.  Corollary 2.5 follows from the estimate (2.6) choosing R = RL as in Corollary 2.3 and if VL = AξnL and WL = AηmL it remains only to verify the assertion that κAξn ,Aηm → ∞ as n, m → ∞ provided that ξ, η ∈ ΩP are not periodic and not shifts of each other. Indeed, π(Aξn ), π(Aηm ) and π(Aξn , Aηm ) are nondecreasing in n and m, and so does π(Aξn , Aηm ). Hence, the limit r = limn,m→∞ κAξn ,Aηm exists. If r < ∞ then, at least, one of the limits r1 = limn→∞ π(Aξn ), r2 = limm→∞ π(Aηm ) or r3 = limn,m→∞ π(Aξn , Aηm ) is finite. If r1 < ∞ then ξ is periodic with the period r1 , if r2 < ∞ then η is periodic with the period r2 and if r3 < ∞ then either T r3 ξ = η or T r3 η = ξ. Finally, it follows from Lemma 3.2 from [16] that (2.12) holds true for P × P -almost all (ξ, η), completing the proof.  References [1] M. Abadi and N. Vergne, Sharp errors for point-wise Poisson approximations in mixing processes, Nonlinearity 21 (2008), 2871–2885. [2] R. Arratia, L. Goldstein and L. 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Yang, Geometric law for numbers of returns until a hazard under φ-mixing, arXiv: 1812.09927. [17] B. Pitskel, Poisson limit law for Markov chains, Ergod. Th. Dynam. Sys. 11 (1991), 501–513. Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel E-mail address: kifer@math.huji.ac.il