On Khinchine type inequalities for pairwise independent Rademacher random variables
aa r X i v : . [ m a t h . F A ] D ec ON KHINTCHINE TYPE INEQUALITIES FOR PAIRWISEINDEPENDENT RADEMACHER RANDOM VARIABLES
BRENDAN PASS AND SUSANNA SPEKTOR
Abstract.
We consider Khintchine type inequalities on the p -th moments ofvectors of N pairwise independent Rademacher random variables. We establishthat an analogue of Khintchine’s inequality cannot hold in this setting with aconstant that is independent of N ; in fact, we prove that the best constant onecan hope for is at least N / − /p . Furthermore, we show that this estimateis sharp for exchangeable vectors when p = 4. As a fortunate consequence ofour work, we obtain similar results for 3-wise independent vectors.2010 Classification: 46B06, 60E15Keywords: Khintchine inequality, Rademacher random variables, k -wise inde-pendent random variables. Introduction
Khintchine’s inequality is a moment inequality with many important applicationsin probability and analysis (see [3, 5, 6, 7, 9] among others). It states that the L p norm of weighted independent Rademacher random variables is controlled by their L norm; a precise statement follows. We say that ε is a Rademacher randomvariable if P ( ε = 1) = P ( ε = −
1) = . Let ε i , i ≤ N , be independent copies of ε and a ∈ R N . Khintchine’s inequality (see e.g. Theorem 2.b.3 in [6] or Theorem12.3.1 in [3]) states that for any p ≥ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ! p ≤ C ( p ) k a k = C ( p ) E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (1)Note that the constant C ( p ) does not depend on N . It is natural to ask whetherthe independence condition can be relaxed; indeed, random vectors with dependentcoordinates arise in many problems in probability and analysis (see e.g. [4] and thereferences therein). In this short paper, we are interested in what can be said whenthe independence assumption on the coordinates is relaxed to pairwise (or, moregenerally, k -wise) independence. Definition 1.1.
We call an N -tuple ε = { ε i } Ni ≥ of Rademacher random variablesa Rademacher vector . For a fixed non-negative integer k , a Rademacher vector iscalled k -wise independent if any subset { ε i , ε i , . . . , ε i k } of length k is mutuallyindependent.When k = 2 in the preceding definition, we will often use the terminology pair-wise independent in place of 2-wise independent. As it will be useful in what follows,we note that instead of random variables, it is equivalent to consider probability BP is pleased to acknowledge the support of a University of Alberta start-up grant andNational Sciences and Engineering Research Council of Canada Discovery Grant number 412779-2012. measures P on the set {− , } N , where P = law ( ε ). The condition that ε is aRademacher vector is then equivalent to the condition that the projections law ( ε i )of P onto each copy of {− , } is equal to P := [ δ − + δ ]. The k -wise indepen-dence condition is equivalent to the condition that the projections law ( ε i , . . . , ε i k )of P onto each k -fold product {− , } k is product measure ⊗ k P .In general, sequences of k -wise independent random variables, for k ≥
2, sharesome of the properties of the mutually independent ones, including the secondBorel-Cantelli lemma and the strong law of large numbers (see e.x. [1]). Otherproperties of mutually independent sequences, such as the central limit theorem,fail to carry over to the pairwise independent setting, however. For more on k -wiseindependent sequences and their construction, see, for example [2, 10, 11].Our goal is to determine whether Khintchine’s inequality holds for k -wise in-dependent Rademacher random variables, and, if not, to understand how badly itfails. More precisely, if we define(2) C ( N, p, k ) = sup a ∈ R N : || a || =1 ε is a k -wise independent Rademacher vector E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ! /p , then the questions we are interested in can be formulated as:1. Is C ( N, p, k ) bounded as N → ∞ , for a fixed p and k ?2. If not, what is the growth rate of C ( N, p, k )?Note that the C ( N, p, k ) form a monotone decreasing sequence in k , as the k -dependence constraint becomes more and more stringent as k increases. We define C ( N, p, ∞ ) to be the best constant in Khintchine’s inequality (for independentrandom variables): C ( N, p, ∞ ) = sup a ∈ R N : || a || =1 E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ¯ ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ! /p . where the ¯ ε i are mutually independent Rademacher random variables. Note that,as mutual independence implies k -wise independent for any k , we have C ( N, p, k ) ≥ C ( N, p, ∞ ). In this notation, the classical Khintchine inequality means that C ( N, p, ∞ )is bounded as N goes to ∞ for each fixed p .Some properties of C ( N, p, k ) are easily discerned. By an application of Holder’sinequality, we get, for any Rademacher ε and any a ,(3) E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ! p ≤ √ N || a || and so we have C ( N, p, k ) ≤ √ N. In fact, for a random vector with law ( ε ) = [ δ , , ,..., + δ − , − , − ,..., − ], we getequality in (3). This ǫ is 1-wise independent, which simply means that it hasRademacher marginals and so we have C ( N, p,
1) = √ N . Clearly, this vector isnot pairwise independent, however, and so it provides no further information on C ( N, p, k ) for k ≥ N KHINTCHINE TYPE INEQUALITIES FOR PAIRWISE INDEPENDENT RADEMACHER RANDOM VARIABLES3
Let us also mention that, when p is an even integer, and k ≥ p , it is actually astraightforward calculation to show that C ( N, p, k ) = C ( N, p, ∞ ) ≈ k (that is, Khintchine’s inequality for k -wise independent random variablesholds with the same constant as in the independence case). This seems to be a”folklore” result, which is well known to experts, but we were unable to find asuitable reference.In this paper, we focus on the k = 2 case. We prove that for p ≥ N even, C ( N, p, ≥ N / − /p , providing a negative answer to the first question above.Moreover, if we define C e ( N, p, k ) as in (2), but with the supremum restricted to exchangeable
Rademacher vectors ε , and consider the p = 4 case, we prove that C e ( N, ,
2) = N / − / = N / .As a fortunate consequence of our work here, we obtain analagous results for k = 3. Understanding the k ≥ Estimates on C ( N, p, k = 2, and provides some information about the second questionin the same case. Theorem 2.1.
Let N = 2 n be even and set a = (1 , , .... ∈ R N . Then, for all p ≥ , (4) sup ε is a k -wise independent Rademacher vector E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ! /p = N / − /p k a k Consequently, C ( N, p, k ) ≥ N / − /p .Proof. We explictly construct a pairwise independent Rademacher vector ε = ( ε , ...ε N )for which we get equality in (4), and then show that this maximizes the left handside over the set of all such vectors. To do this, we will define the probabilitymeasure P = law ( ε ).We define P = N P a + N − N P b where P a = [ δ , , ,..., + δ − , − , − ,..., − ] is uniformmeasure on the two points (1 , , , . . . , , ( − , − , . . . , − ∈ {− , } N and P b isuniform measure on the set of all points with an equal number of 1’s and − { , , . . . , | {z } N/ of them − , − , . . . , − | {z } N/ of them } . We first verifythat this is pairwise independent probability measure; that is, that it’s twofoldmarginals are ( δ , + δ , − + δ − , + δ − , − ). By symmetry between the coordinates,it suffice to verify this fact for the projection P on the first two copies of {− , } .To see this, we have P (1 ,
1) = 1
N P a (1 , , , . . . ,
1) + N − N P b { ε : ( ε , ε ) = (1 , } . Now, P a (1 , , , . . . ,
1) = , and it is easy to see that P b { ε : ( ε , ε ) = (1 , } = N/ − N − , implying P (1 ,
1) = 12 N + ( N − N/ − N ( N −
1) = 14 . BRENDAN PASS AND SUSANNA SPEKTOR
Similar calculations imply P (1 , −
1) = P ( − ,
1) = P ( − , −
1) = , and so P ispairwise independent.Now, letting ε = ( ε , ...ε N ) be a random variable with law ( ε ) = P , and notingthat | P Ni =1 ε i | p is 0 for points in the support of P b and N for points in the supportof P a , we have E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = 1 N N p = N p − Noting that || a || = √ N , it follows that " E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p /p = N − /p = √ N N / − /p = || a || N / − /p It remains to show that the ε we have constructed is optimal in (4) . Define thefunctions u : { , − } → R by u (1 ,
1) = u ( − , −
1) = N p / (cid:18) N (cid:19) and u (1 , −
1) = u ( − ,
1) = − N − N u (1 ,
1) = − ( N − N p − (cid:0) N (cid:1) . We will show that for any ( ε , . . . , ε N ) ∈ { , − } N , we have(5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ N X i 1) + (cid:18) N − n (cid:19) u ( − , − 1) + n ( N − n ) u (1 , − | n − N | p ≤ (cid:20)(cid:18) n (cid:19) + (cid:18) N − n (cid:19) − n ( N − n ) N − N (cid:21) u (1 , (cid:20) n ( n − N − n )( N − n − − n ( N − n ) N − N (cid:21) N p N ( N − (cid:12)(cid:12)(cid:12)(cid:12) n − NN (cid:12)(cid:12)(cid:12)(cid:12) p ≤ (cid:20) n ( n − N − n )( N − n − − n ( N − n ) N − N (cid:21) N ( N − (cid:12)(cid:12)(cid:12)(cid:12) n − NN (cid:12)(cid:12)(cid:12)(cid:12) . As clearly 0 ≤ n ≤ N , we have that (cid:12)(cid:12)(cid:12)(cid:12) n − NN (cid:12)(cid:12)(cid:12)(cid:12) ≤ 1. As p ≥ 2, we N KHINTCHINE TYPE INEQUALITIES FOR PAIRWISE INDEPENDENT RADEMACHER RANDOM VARIABLES5 then clearly have (cid:12)(cid:12)(cid:12)(cid:12) n − NN (cid:12)(cid:12)(cid:12)(cid:12) p ≤ (cid:12)(cid:12)(cid:12)(cid:12) n − NN (cid:12)(cid:12)(cid:12)(cid:12) , with equality only when 2 n − N = 0 , N or − N . That is, we have equality precisely when n = 0 , N or N , which correspondexactly to points in the support of P . This establishes (5), with equality only when ǫ is in the support of P .Now note that by (5) for any pairwise independent Rademacher vector ˜ ε on { , − } N , we have E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ˜ ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ E (cid:0) N X i ≤ j u (˜ ε i , ˜ ε j ) (cid:1) = N X i ≤ j E (cid:0) u (˜ ε i , ˜ ε j ) (cid:1) But the right hand side is constant on the set of pairwise independent Rademachervectors, as it depends only on the twofold vectors (˜ ε i , ˜ ε j ). As we have equality in(5) P almost surely, for the particular sequence ε with law ( ε ) = P , we get E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = E (cid:0) u ( ε i , ε j ) (cid:1) , and therefore, E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ˜ ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p completing the proof. (cid:3) Remark 2.2. It is worth mentioning that the linear program (4) is reminiscentof a discrete version of the multi-marginal optimal transport problem (see [8] andthe references therein). The difference is that here, the twofold marginals P = law ( ε i , ε j ) are prescribed, rather than the marginal P = law ( ε i ). Moreover, thefunction u that shows up in the proof can be interpreted in terms of the dualprogram, which is is to minimize E X i = j u ij ( ε i , ε j ) over all collections of functions u ij : {− , } → R , for 1 ≤ i = j ≤ N , which satisfythe constraint X i = j u ij ( ε i , ε j ) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p for all ε = ( ε , . . . , ε N ) ∈ {− , } N . The minimizing functions for this dual programare exactly u ij = u for all i, j , where u is as in the proof above.Next, we turn our attention to the second question in the introduction, andtry to determine the precise behaviour of C ( N, p.k ). For this, we study only the k = 2 , p = 4 case, and restrict our attention to exchangeable Rademacher vectors. Lemma 2.3. For any a ∈ R N with || a || = 1 and any pairwise independent,exchangeable ε , we have BRENDAN PASS AND SUSANNA SPEKTOR E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 √ N ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ¯ ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) where ¯ ε denotes an independent Rademacher vector.Proof. We assume, without loss of generality, that a i ≥ i . After expand-ing the power, and noting that the Rademacher condition implies ε i = 1 almostsurely, the pairwise independence condition implies E ( ε i ε j ε k ) = E ( ε i ε j ) = E (¯ ε i ¯ ε j ) = E (¯ ε i ¯ ε j ¯ ε k ) E ( ε i ε j ) = E (¯ ε i ¯ ε j ) E ( ε i ε j ) = E (¯ ε i ¯ ε j ) E ( ε i ) = E (¯ ε i ) . Therefore, we have(6) E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ¯ ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X i,j,k,l distinct a i a j a k a l E ( ε i ε j ε k ε l )By exchangability, E ( ε i ε j ε k ε l ) is independent of i, j, k and l ; denoting E ( ε i ε j ε k ε l ) := c , this gives us(7) E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ! = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ¯ ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ! + c X i,j,k,l distinct a i a j a k a l . Now, if c ≤ 0, the above is less than E (cid:16)(cid:12)(cid:12)(cid:12)P Ni =1 a i ¯ ε i (cid:12)(cid:12)(cid:12) p (cid:17) , and the proof is complete.If, on the other hand, c ≥ 0, we have, by Maclaurin’s inequality, X i,j,k,l distinct a i a j a k a l ≤ (cid:18) N (cid:19) (cid:0) N (cid:1) X i = j a i a j ≤ (cid:18) N (cid:19) (cid:0) N (cid:1) X i = j ( a i + a j ) = 24 (cid:18) N (cid:19) " (cid:0) N (cid:1) N − || a || = 24 (cid:18) N (cid:19) N . (8)We have equality when all of the a i = √ N . Note that this implies that(9) E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 √ N ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 √ N ¯ ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + c (cid:18) N (cid:19) N 2N KHINTCHINE TYPE INEQUALITIES FOR PAIRWISE INDEPENDENT RADEMACHER RANDOM VARIABLES7 Now, it is well known that E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ¯ ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 √ N ¯ ǫ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and so plug-ging this and (8) into (7), we have E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 √ N ¯ ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + c (cid:18) N (cid:19) N (10) = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 √ N ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (11)where the last line follows from (9). This completes the proof. (cid:3) Theorem 2.1, the classical Khintchine inequality, and the preceding lemma nowimply the following variant of Khintchine’s inequality, for p = 4 and pairwise inde-pendent, exchangeable Rademacher random vectors. Corollary 2.4. We have C e ( N, , 2) = N / . In particular, for any a ∈ R N andany pairwise independent exchangeable Rademacher vector ǫ , we have E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 a i ε i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N || a || . Remark 2.5. It is straightforward to verify that the measure P = law ( ε ) derivedin Theorem is in fact 3-wise independent, and thus we immediately obtain analoguesof the preceding results for k = 3: C ( N, p, ≥ N / − /p and C e ( N, , 3) = N / . Remark 2.6. Generally speaking, one can identify exchangeable, k -wise indepen-dent random variables x , . . . , x N on R having equal fixed marginals P = law ( x i )with permutation symmetric probability measures P = law ( x , . . . , x N ) on R N whose k -fold marginals are ⊗ k P . The set of measures satifying these constraintsis a convex set, and identifying the set of extremal points, or vertices, of this set isan interesting and nontrivial question.It is easy to see upon inspection of the proof of Theorem 2.1 that the mea-sure law ( ε ) we construct is the unique maximizer of the linear functional law ( ε ) E ( P Ni =1 ε i ) on the convex set of symmetric, pairwise independent Rademacher prob-ability measures. As a consequence of this proof, we have therefore identified anextremal point of this set. References [1] D. Andrews, Laws of Large Numbers for Dependent Non-Identically Distributed RandomVariables Econometric Theory, , (1988), 458–467.[2] Y. Derriennic, A. Klopotowski, Cinq variables al ´ eatoires, binaires, Institut Galil´ee, UniversiteParis XIII, (1991), 1–38. (1985), 109–117. BRENDAN PASS AND SUSANNA SPEKTOR [3] D. J. H. Garling, Inequalities: A Journey into Linear Analysis , Cambridge University Press,Cambridge, 2007.[4] O. Guedon, P. Nayar, T. Tkocz, Concentration inequalities and geometry of convex bod-ies , Extended notes of a course, Polish Academy of Sciences of Warsaw, to appear.(http://perso-math.univ-mlv.fr/users/guedon.olivier/listepub.html)[5] J.P. Kahane, Some random series of functions , Second edition. Cambridge Studies in Ad-vanced Mathematics, 5, Cambridge University Press, Cambridge, 1985.[6] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I and II, Springer, 1996.[7] V. D. Milman, G. Schechtman, Asymptotic theory of finite-dimensional normed spaces. Withan appendix by M. Gromov. Lecture Notes in Math., 1200. Springer-Verlag, Berlin, 1986.[8] B. Pass Multi-marginal optimal transport: theory and applications. To appear in ESAIM:Math. Model. Numer. Anal.[9] G. Peskir, A. N. Shiryaev, The inequalities of Khintchine and expanding sphere of theiraction, Russian MAth. Surveys Independence and fair coin-tossing, Math. Scientist, (1985), 109–117.[11] J. Robertson, A two state pairwise independent stationary process for which x x x is de-pendent, Sankhy¯a, Series A, , (1988), 171–183. (1985), 109–117. Brendan Pass Address: University of Alberta, Edmonton, AB, Canada, T6G2G1 E-mail address : [email protected] Susanna Spektor Address: Michigan State University, East Lansing, MI, USA, 48824 E-mail address ::