On permutation-invariance of limit theorems

Transcription

1 On permutation-invariance of limit theorems I. Beres an R. Tichy Abstract By a classical principle of probability theory, sufficiently thin subsequences of general sequences of ranom variables behave lie i.i.. sequences. This observation not only explains the remarable properties of lacunary trigonometric series, but also provies a powerful tool in many areas of analysis, such the theory of orthogonal series an Banach space theory. In contrast to i.i.. sequences, however, the probabilistic structure of lacunary sequences is not permutation-invariant an the analytic properties of such sequences can change after rearrangement. In a previous paper we showe that permutation-invariance of subsequences of the trigonometric system an relate function systems is connecte with Diophantine properties of the inex sequence. In this paper we will stuy permutation-invariance of subsequences of general r.v. sequences. AMS 2000 Subject classification. Primary 42A55, 42A6, 60F05, 60G09. Key wors an phrases: lacunary series, limit theorems, permutation-invariance, subsequence principle, exchangeable sequences Graz University of Technology, Institute of Statistics, Koperniusgasse 24, 800 Graz, Austria. beres@tugraz.at. Research supporte by FWF grants P24302-N8, W230 an OTKA grant K Graz University of Technology, Institute of Mathematics A, Steyrergasse 30, 800 Graz, Austria. tichy@tugraz.at. Research supporte by FWF grants P24302-N8 an W230.

2 Introuction It is nown that sufficiently thin subsequences of general r.v. sequences behave lie i.i.. sequences. For example, Révész [23] showe that if a sequence (X n ) of r.v. s satisfies sup n EXn 2 <, then one can fin a subsequence (X n ) an a r.v. X L 2 such that = c (X n X) converges a.s. provie = c2 <. Uner the same conition, Gaposhin [3], [4] an Chatterji [9], [0] prove that there exists a subsequence (X n ) an r.v. s X L 2, Y L, Y 0 such that an lim sup N (X n X) N(0, Y ) (.) N N (X n X) = Y /2 a.s.. (.2) 2N log log N N Here N(0, Y ) enotes the istribution of the r.v. Y /2 ζ, where ζ is a stanar normal r.v. inepenent of Y. Komlós [8] showe that if sup n E X n <, then there exists a subsequence (X n ) an a r.v. X L such that lim N N N X n = X = Chatterji [8] showe that if sup n E X n p < where 0 < p < 2, then the conclusion of the previous theorem can be change to lim N N /p a.s.. N (X n X) = 0 a.s. = for some X L p. Note the ranomization in all these examples: the role of the mean an variance of the subsequence (X n ) is playe by ranom variables X, Y. For further limit theorems for subsequences of general r.v. sequences an for the history of the topic until 966, see Gaposhin [3]. Since the asymptotic properties of an i.i.. sequence o not change if we permute its terms, it is natural to expect that limit theorems for lacunary subsequences of general r.v. sequences remain vali after any permutation of their terms. This is, however, not the case. By classical results of Salem an Zygmun [24], [25] an Erős an Gál [2], uner the Haamar gap conition the sequence (sin 2πn x) satisfies n + /n q > =, 2,... (.3) an lim sup N N sin 2πn x N(0, ) (.4) N/2 = N log log N N = sin 2πn x = a.s. (.5) 2

3 with respect to the probability space ((0, ), B, µ), where µ enotes the Lebesgue measure. Erős [] an Taahashi [27] prove that (.4), (.5) remain vali uner the weaer gap conition n + /n + c α, =, 2,... (.6) for 0 < α < /2 an that for α = /2 this becomes false. As it was shown in [2], [3], uner the Haamar gap conition (.3) the CLT (.4) an the LIL (.5) are permutation-invariant, i.e. they remain vali after any permutation of the sequence (n ), but this generally fails uner the gap conition (.6). Similar results hol for lacunary sequences f(n x), where f is a measurable function satisfying f(x + ) = f(x), 0 f(x) x = 0, 0 f 2 (x) x <. (.7) In this case, assuming the Haamar gap conition (.3), the valiity of the CLT N N = f(n x) N(0, σ 2 ) (.8) an of its permute version epen on the number of solutions of the Diophantine equation an + bn l = c,, l N. (.9) As shown in [], [2], [3], a sharp conition for the CLT is that the number of solutions of (.9) is o(n) for any fixe nonzero a, b, c, while the permute CLT requires the stronger boun O() for the number of solutions. Permutation-invariance of limit theorems becomes a particularly ifficult problem for parametric limit theorems, e.g. for limit theorems containing arbitrary coefficients. By a classical result of Menshov [20], from every orthonormal system (f n ) one can select a subsequence (f n ) which is a convergence system, i.e. the series = c f n converges almost everywhere provie = c2 <. The question of whether a subsequence (f n ) exists such that this property remains vali after any permutation of (f n ) (i.e., by the stanar terminology, (f n ) is an unconitional convergence system) remaine open for nearly 40 years until it was answere in the affirmative by Komlós [9]. For another proof see Alous [4]. The problem of whether every orthonormal system can be rearrange to become a convergence system is still open; for a partial result see Garsia [5]. Kolmogorov showe (see [7]) that there exists an f L 2 (0, ) whose Fourier series, suitably permute, iverges a.e. But even though the Raemacher-Menshov convergence theorem yiels a sharp a.e. convergence criterion for orthonormal series, there is no similar complete result for rearrange trigonometric series. The previous results show that permutation-invariance of limit theorems lies substantially eeper than that of the original theorems an raise the question of which limit theorems hol in a permutation-invariant form for lacunary sequences. In this paper we will prove the surprising fact that, in a sense to be mae precise, all nonparametric istributional limit theorems for i.i.. ranom variables hol for lacunary subsequences (f n ) of general r.v. sequences in a permutation-invariant form provie 3

4 that the subsequence is sufficiently thin, i.e. the gaps of the sequence (epening on the limit theorem) grow sufficiently rapily. We will euce this result from a general structure theorem for lacunary sequences prove in [6] stating that sufficiently thin subsequences of any tight sequence of ranom variables are nearly exchangeable. While this iea is simple an elementary, formulating our results is somewhat technical an requires some preparations in Section 2. The proof of our theorem will be given in Section 3. 2 Main result We start with a formal efinition of the concept wea limit theorem. Let M enote the set of all probability measures on R an ϱ the Prohorov metric on M efine by Here ϱ(ν, λ) = inf { ε > 0 : ν(a) λ(a ε ) + ε an λ(a) ν(a ε ) + ε for all Borel sets A R }. A ε = {x R : x y < ε for some y A} enotes the open ε-neighborhoo of A. A ranom measure is a measurable map from a probability space to M. The following efinition is ue to Alous [4]. Definition. A wea limit theorem of i.i.. ranom variables is a system where (a) S is a Borel subset of M; T = (f, f 2,..., S, {G µ, µ S}) (b) For each, f = f (x, x 2,..., µ) is a continuous function on R M, satisfying the Lipschitz conition f (x, x 2,..., µ) f (x, x 2,..., µ) where 0 c,i an lim c,i = 0 for all i; c,i x i x i (c) For each µ S, G µ is a probability istribution on R such that the function µ G µ is measurable (with respect to the Borel σ-fiels in S an M); an () If µ S an X, X 2,... are inepenent r.v. s with common istribution µ then f (X, X 2,..., µ) i= G µ as. (2.) For example, the central limit theorem correspons to S = {µ M : x 2 µ(x) < }, G µ = N(0, Var µ), 4

5 f (x, x 2,..., µ) = (x x Eµ)/, c,i = /2 I {i }. The theorem itself is expresse by (2.). Using the terminology of [7], we call a sequence (X n ) of ranom variables etermining if it has a limit istribution relative to any set A in the probability space with P (A) > 0, i.e. for any A Ω with P (A) > 0 there exists a istribution function F A such that lim n P (X n t A) = F A (t) for all continuity points t of F A. Here P ( A) enotes conitional probability given A. (This concept is the same as that of stable convergence, introuce by Rényi [22]; our terminology follows that of functional analysis.) By an extension of the Helly-Bray theorem (see [7]), every tight sequence of r.v. s contains a etermining subsequence. As is shown in [4], [7], for any etermining sequence (X n ) there exists a ranom measure µ (i.e. a measurable map from the unerlying probability space (Ω, F, P) to M) such that for any A with P (A) > 0 an any continuity point t of F A we have F A (t) = E A ( µ(, t]) (2.2) where E A enotes conitional expectation given A. We call µ the limit ranom measure of (X n ). The following result is Alous celebrate subsequence theorem [4]. Theorem 2. Let (X n ) be a etermining sequence with limit ranom measure µ. Let T = (f, f 2,..., S, {G µ, µ S}) be a wea limit theorem an assume P ( µ S) =. Then there exists a subsequence (X n ) such that f (X n, X n2,..., µ) G µ P. (2.3) In case of the CLT formalize above, assuming sup n EX 2 n < + implies easily that µ has finite variance almost surely an thus enoting its mean an variance by X an Y, respectively, we see that the integral in (2.3) is the istribution N(0, Y ). Hence (2.3) states in the present case that N N = (X n X) N(0, Y ) which is exactly the CLT of Chatterji [9] an Gaposhin [4] formulate in the Introuction. Theorem 2. shows that a similar subsequence theorem hols for any wea limit theorem of i.i.. ranom variables. For a version of this result for strong (a.s.) limit theorems, we refer to Alous [4]. In what follows, we change the technical conitions on f in the efinition of wea limit theorems slightly, leaing to a class more convenient for our purposes. Definition. The limit theorem T = (f, f 2,..., S, {G µ, µ S}) is calle regular if there exist two sequences p q of positive integers tening to + an a sequence ω + such that 5

6 (i) f (x, x 2,..., µ) epens only on x p,..., x q, µ (ii) f satisfies the Lipschitz conition f (x p,..., x q, µ) f (x p,..., x q, µ ) ω q i=p x i x i α + ϱ (µ, µ ) (2.4) for some 0 < α where ϱ is a metric on S generating the same topology as the Prohorov metric ϱ. Thus in this case the function f epens only on a finite segment x p,... x q of the variables x, x 2,.... On the role of ϱ see [4]. The above efinition brings out clearly the crucial feature of limit theorems, namely the fact that the valiity of the theorem oes not epen on finitely many terms of (X n ), while the original efinition assumes only that the epenence of f (X, X 2,...) on any fixe variable X j of the sequence is wea if is large. However, there is very little ifference between these assumptions. For example, the central limit theorem can be formalize by either of the functions f (x,..., x, µ) = (x x Eµ)/ an f (x [ /4 ],..., x, µ) = (x [ /4 ] x Eµ)/ of which the secon leas to a regular limit theorem with the Wasserstein metric ( /2 ϱ (µ, µ ) = Fµ (x) F µ (x) x) 2, 0 where F µ, F µ enote the istribution function of µ an µ, respectively. Uner boune secon moments, the contribution of the first /4 terms in the norme sum efining f are irrelevant an thus we can always switch from f to f an bac again. The same proceure applies in the general case. We are now in a position to formulate the main result of our paper. Theorem 2.2 Let (X n ) be a etermining sequence with limit ranom measure µ. Let T = (f, f 2,..., S, {G µ, µ S}) be a regular wea limit theorem an assume that P ( µ S) =. Then there exists a subsequence (X n ) = (Y ) such that for any permutation (Y ) of (Y ) we have f (Y, Y 2,..., µ) G µ P. (2.5) Note that we assume the regularity of the limit theorem, but as we pointe out before, this is no restriction of generality. The limit theorem T in Theorem 2.2 is nonparametric, i.e. the function f epens on x, x 2,... an µ, but on no aitional parameters. A simple example of a parametric istributional limit theorem is the weighte CLT, where f = A a j (x j Eµ), A = j= 6 j= a 2 j /2.

7 For any fixe coefficient sequence (a ) this efines a nonparametric limit theorem T an Theorem 2.2 applies, but the selecte subsequence (X n ) epens on (a ). As the iscussion above shows, in the case of a parametric limit theorem T eciing whether a universal subsequence (X n ) woring for all parameters is generally a very ifficult problem; an example of a limit theorem where such a choice is impossible is given in [6]. For this reason, in the present paper we eal only with nonparametric limit theorems. In Alous [4] a formalization of strong limit theorems is also given an the analogue of Theorem 2. is prove. Using a reformulation of strong limit theorems as a sequence of probability inequalities as given in [5], [6], a version of our Theorem 2.2 can be given for a subclass of limit theorems consiere in [4]. We also mention that for a more limite class of wea limit theorems Theorem 2.2 was prove in [2]. 3 Proof of Theorem 2.2. To simplify the formulas, let f (µ) enote, for any µ S, the istribution of the ranom variable f (ξ, ξ 2,..., µ) where ξ, ξ 2,... are inepenent r.v. s with common istribution µ. The following statements are easy to verify: (A) If ϱ(µ, ν) ε then ϱ(f (µ), f (ν)) ε α q + ϱ (µ, ν) where α, q an ϱ are the quantities appearing in (2.4). (B) Let µ,..., µ r an ν,... ν r be probability istributions, further let c,..., c r be nonnegative numbers with r i= c i =. Assume that the sum of those c i s such that ϱ(µ i, ν i ) ε is at most ε. Then the Prohorov istance between r c i ν i is at most 2ε. i= r i= c i µ i an (C) Let µ an ν be ranom measures (i.e. measurable maps from a probability space (Ω, F, P) to M) such that P (ϱ( µ, ν) ε) ε. Then the Prohorov istance between µp an νp is 2ε. To prove statement (A) note that if ϱ(µ, ν) ε then by a theorem of Strassen [26] there exist, on some probability space, r.v. s ξ an η with istribution µ an ν such that P ( ξ η ε) ε. On a larger probability space, let (ξ n, η n ) (n =, 2,...) be inepenent ranom vectors istribute as (ξ, η). Clearly P ( ξ i η i ε) ε (i =, 2,...) an thus using (2.4) we see that f (ξ p,..., ξ q, µ) f (η p,..., η q, ν) ε α q + ϱ (µ, ν) except on a set with probability εq ε α q, proving (A). (Clearly we can assume 0 < ε an that in the efinition of regular limit theorems we have ω for all.) Statements (B) an (C) are almost evient, (B) is a special case of (C). To prove our theorem, let (X n ) be a etermining sequence of r.v. s with limit ranom measure µ. Then (X n ) is tight, i.e. sup j P ( X j t) 0 as t. As ω +, we can choose a nonecreasing sequence (r ) of integers tening to + so slowly that r min(p, ω /4 ) (3.) 7

8 an ( sup P X j ) j 2 ω/(4α) Let (ε ) ten to 0 monotonically an so rapily that 2 r 2 ( ). (3.2) ε α r q. (3.3) Using the structure theorem [6, Theorem 2], it follows that there exists a subsequence (X n ) an a sequence (X ) of r.v. s such that X n X = O(2 ) a.s. (3.4) an X has the following properties: (A ) Each X taes only finitely many values (B ) σ{x } σ{x 2 }... (C ) For each the atoms of the finite σ-fiel σ{x r } can be ivie into two classes Γ an Γ 2 so that A Γ P (A) ε r (3.5) an for any A Γ 2 there exist i.i..r.v. s {Z (A) j, j = r +, r + 2,...} efine on A with istribution function F A such that P A ( X j Z (A) j ε r ) εr j = r +, r + 2,.... (3.6) Here F A enotes the limit istribution of (X n ) on the set A (which exists since (X n ) is etermining) an P A enotes conitional probability with respect to A. Let µ n enote the ranom measure efine by µ n (B) = E( µ(b) X n). By Lemma 7 of [6] we have µ n µ a.s. an thus by passing to a further subsequence of (X n ) we can also assume that P {ϱ( µ n, µ) ε n } ε n (3.7) P {ϱ ( µ n, µ) ε n } ε n. (3.8) We show that the last obtaine subsequence (X n ) satisfies the conclusion of the theorem. In view of (2.4) an (3.4), X n an X are interchangeable in the statement of the theorem an thus it suffices to prove that if (X ) satisfies statements (A ), (B ), (C ) above then for any permutation (Y ) of (X ) we have (2.5). To verify this, note that by (2.4) an (3.6) we have { P A f (X i,..., X i l, µ A ) f (Z (A) i,..., Z (A) i l, µ A ) ε α } r q (3.9) ε α r q A Γ 2 where l = q p +, i,..., i l are ifferent integers > r an µ A is the probability measure corresponing to F A. (Note that we o not assume here i <... < i l ; the vectors (X i..., X i l ) an (Z (A) i,..., Z (A) i l ) are close to each other coorinatewise, i.e. for any orer of i,..., i l. Since the Z (A) j are i.i.., the istribution of the 8

9 vector (Z (A) i,..., Z (A) i l ) is permutation-invariant, proviing an explanation for the phenomenon escribe in Theorem 2.2.) Since (3.9) is vali for all A Γ 2 an µ A in (3.9) is ientical to µ r on A (see Lemma 6 of [5]), using (3.5), (3.9) an statement (B) at the beginning of the proof we get ϱ ( f (X i,..., X i l, µ r ), A f (µ A )P (A) ) 2ε α r q (3.0) where the sum is extene for all atoms A of σ{x r } an a r.v. in a Prohorov istance is meant as its istribution. Next we show that (3.0) remains vali, with the right han sie increase by r, if i,..., i l, l = q p +, are arbitrary ifferent positive integers (not necessarily > r ). Inee, remove from X i,..., X i l those whose inex is r an replace them with (ifferent) X j s with j > max(r, i,..., i l ). This means that we change f (X i,..., X i l, µ r ) at most at r locations an at each such position we replace an X µ by an X ν where µ r an ν > r. By (2.4), f changes at most by X ω µ X ν α =: W where the sum has r terms. Using (3.), (3.2) we get ( ) P ( W r ) P ( W ω /2 ) = P X µ X ν α ω /2 ( P X µ X ν ( /2 ω ) /α) ( 2r sup r j P X j 2 ω/(4α) ) r an thus the above changes increase the left han sie of (3.0) by at most r, i.e. ( ϱ f (X i,..., X i l, µ r ), ) f (µ A )P (A) 2ε α r q + r (3.) A for any ifferent positive integers i,..., i l, l = q p +. Changing µ r into µ will change f (X i,..., X i l, µ r ) on the left han sie of (3.) by at most ε r, except on a set of probability ε r (see (3.8) an (2.4)) an thus the left han sie of (3.) changes by at most ε r. Thus observing that the sum A f (µ A )P (A) in (3.) equals f ( µ r )P, we prove the following Proposition. Let (X ) be any permutation of (X ). Then ϱ ( f (Xp,..., Xq, µ), f ( µ r )P ) 3ε α r q + r. To complete the proof of our theorem it suffices to show that the Prohorov istance of any two of the istributions f ( µ r )P f ( µ)p G µ P (3.2) tens to zero as. To verify this observe first that (3.7), (3.8) an statement (A) at the beginning of the proof imply that the Prohorov istance of f ( µ r ) an 9

10 f ( µ) is ε α r q + ε r, except on a set with probability ε α r q + ε r an thus by statement (C) an (2.2) the Prohorov istance of the first two istributions in (3.2) is 2(ε α r q + ε r ) 4. On the other han, the valiity of f (µ) G µ for any µ S (which is a part of the efinition of a wea limit theorem) an P ( µ S) = imply ϱ(f ( µ), G µ ) 0 a.s. an thus there exists a numerical sequence δ 0 such that P {ϱ(f ( µ), G µ ) δ } δ ( =, 2,...). Thus by statement (C) above we get that the Prohorov istance of the secon an thir istribution in (3.2) is 2δ. This completes the proof of Theorem 2.2. Acnowlement. We woul lie to than two anonymous referees for their remars leaing to a substantial improvement of the presentation. References [] C. Aistleitner an I. Beres. On the central limit theorem for f(n x). Probab. Theory Rel. Fiels 46 (200), [2] C. Aistleitner, I. Beres an R. Tichy. On the law of the iterate logarithm for permute lacunary sequences. Proc. Stelov Inst. Math. 276 (202), [3] C. Aistleitner, I. Beres an R. Tichy. On permutations of lacunary series. RIMS Kôyûrou Bessatsu B34 (202), 25. [4] D. J. Alous. Limit theorems for subsequences of arbitrarily-epenent sequences of ranom variables, Z. Wahrscheinlicheitstheorie verw. Gebiete 40 (977), [5] I. Beres. An extension of the Komlós subsequence theorem. Acta Math. Hung. 55 (990) [6] I. Beres an E. Péter. Exchangeable ranom variables an the subsequence principle, Prob. Theory Rel. Fiels 73 (986), [7] I. Beres an H. P. Rosenthal. Almost exchangeable sequences of ranom variables, Z. Wahrscheinlicheitstheorie verw. Gebiete 70 (985), [8] S. D. Chatterji. A general strong law. Invent. Math / [9] S. D. Chatterji. A principle of subsequences in probability theory: The central limit theorem. Av. Math. 3 (974), [0] S. D. Chatterji. A subsequence principle in probability theory II. The law of the iterate logarithm. Invent. Math. 25 (974), pp Springer, 972. [] P. Erős. On trigonometric sums with gaps. Magyar Tu. Aa. Mat. Kut. Int. Közl. 7 (962),

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