Khinchine inequality for slightly dependent random variables

Transcription

1 arxiv: v1 [mathpr] 7 Aug 017 Khinchine inequality for slightly deendent random variables Susanna Sektor Abstract We rove a Khintchine tye inequality under the assumtion that the sum of Rademacher random variables equals zero As an alication we show a new tail-bound for a hyergeometric random variable 010 Classification: 46B06, 60E15 Keywords: Khintchine inequality, Kahane inequality, Rademacher random variables, Hyergeometric distribution 1 Introduction The Khinchine inequality lays a crucial role in many dee results of Probability and Analysis (see [6, 9, 10, 1, 14, 18] among others) It says that L and L norms of sums of weighted indeendent Rademacher random variables are comarible More recisely, we say that ε 0 is a Rademacher random variableif P(ε 0 = 1) = P(ε 0 = 1) = 1 Let ε i, i N, be indeendent coies of ε 0 and a R N The Khinchine inequality (see eg Theorem b3 in [10] or Theorem 131 in [6]) states that for any one has ( E a i ε i ) 1 a = E a i ε i 1 (1) Note that the (Rademacher) random vector ε = (ε 1,,ε N ) in the Khinchine inequality has indeendent coordinates However in many roblems of Analysis and Probability it is imortant to consider random vectors with deendent coordinates, eg so-called log-concave random vectors, which in general have deendent coordinates, but whose behaviour is similar to that 1

2 of Rademacher random vector or to the Gaussian random vector (see eg [7] and references there in) In [13] the author considered random matrices, whose rows are indeendent random vectors satisfying certain conditions (so the vectors may have deendent coordinates) He studied limiting emirical distribution of eigenvalues of such matrices As an examle of such a vector, showing that the conditions cover large class of natural distributions, not covered by reviously known results, O Rourke considered the vector ε = (ε 1,,ε N ), whose coordinates are Rademacher random variables under the additional condition S = ε i = 0 () (see Examles 14 and 110 in [13]) For such vectors he roved a Khintchine tye inequality with the factor C N/logN in front of a, which was enough for his uroses The goal of this aer is to show that such random variables satisfy a Khintchine tye inequality with the same factor as in the standardkhintchine inequality To shorten notation, by E S we denote an exectation with assumtion () Note that the corresonding robability sace is { } Ω = ε { 1,1} N ε i = 0 = { ε { 1,1} N card{i : ε i = 1} = n } Our main result is the following theorem Theorem 11 Let ε i,i N, be Rademacher random variables satisfying condition () Let a = (a 1,,a N ) R N and b = 1 a N i Then ( N ) 1/ E S a i ε i ( a N b) 1/ N E S a i ε i The first ste in the roof is a reformulation in terms of random variables on the ermutation grou as follows Let N = n For the set Ω defined in 1/ (3) (4)

3 (3), we ut into corresondence the grou Π N of all ermutations of the set {1,,N} as σ Π N A σ = {ε Ω ε i = 1 if σ(i) n; ε i = 1 if σ(i) > n} Given a R N, define f a : Π N R by n f a (σ) := a σ(i) n i=n+1 a σ(i) (5) ByE Π wedenotetheaverageoverπ N, ie theexectationwithresecttothe normalized counting measure on Π N Note, that E S N a iε i = EΠ f Therefore Theorem 11 is equivalent to the following theorem Theorem 1 Let N = n, a R N Let f a be the function defined in (5) Let b = 1 a N i Then, for (E Π f a ) 1/ ( N ) 1/ a i N b ( E Π f a ) 1/ (6) In Section we rove Theorem 1 Then, in Section 3, we consider a secial case of our roblem, when the coordinates of the vector a are either ones or zeros This articular case leads to the hyergeometric distribution We obtain a new bounds for the -th central moments of such variables Finally let us note the setting of Theorem 1 can be extended to more general case We ose the following roblem Problem13 Leta,b R N EstimateM(a,b,) as afunction of a,b,,n,m(a,b,) For examle, is it true that M(a,b,) M(a,b,)+C(a,b,,N), or that M(a,b,) C(,N)M(a,b,)? We discuss this roblem in the last Section 3

4 Proof of Theorem 1 Direct calculations show that E Π f = ) ( N a N a i N(N 1) Thus, without loss of generality we may assume that a i = 0 Fork ndenoteb k,σ := a σ(k) a σ(n+k) andbyh k,σ := (with H n,σ = 0) Clearly, n a σ(i) i=k+1 n a σ(i) i=n+k+1 n a σ(i) n i=n+1 a σ(i) = b 1,σ +H 1,σ = b 1,σ +b,σ +H,σ = = n b i,σ Note, that E Π b 1,σ +H 1,σ = E Π b 1,σ +H 1,σ Hence, n E Π f a (σ) = E Π a σ(i) n i=n+1 a σ(i) = E Π b 1,σ +H 1,σ +E Π b 1,σ +H 1,σ Thus, denoting by δ i,i n, iid Rademacher random variables indeendent of ε 1,,ε N, and using Khinchine inequality (1), we obtain E Π f a (σ) = E Π E δ1 δ 1 b 1,σ +H 1,σ n = E Π E δ1 E δ δ 1 b 1,σ +δ b,σ +H,σ = = E Π E δ1 E δ E δn δ i b i,σ E Π / E Π ( ( n n which comletes the roof b i,σ ) 1/ = / E Π ( n ) / aσ(i) a σ(i+n) ( ) ) / a σ(i) +a σ(i+n) () / a, 4

5 3 Hyergeometric distribution In this section we discuss a secific case of hyergeometric distribution and show how it is related to our roblem Recall that hyergeometric random variable with arameters (N,n,l) is a random variable ξ which takes values k = 0,,l with robability ( l N l ) k = k)( n k ( N n) In this section we consider only the case N = n, l n It is well known that Eξ = l/ In the next roosition we estimate the central moment of ξ Proosition 31 Let 1 l n Let ξ be (n,n,l) hyergeometric random variable Then for one has E ξ Eξ ( ) l 4 Remark 3 It is well known (see eg Theorem 115 in [5]) that the conclusionofproosition31isequivalent tothefollowing, so-called ψ deviation inequality ( ) ct t 1 P( ξ Eξ > t) ex, (7) l in the following sense: Proosition 31 imlies (7) for some absolute constant c > 0 and vise versa, there is another absolute constant C > 0, such that 7 imlies Proosition 31 This estimate, for hyergeometric ξ, is of indeendent interest, in articular it is better than the reviously observed bound ex( t /n) when l n (see Section 65 of [8] and formulas (10), (14) in [16]) Remark 33 One can use Theorem 1 to estimate E S n a iε i in the case that the vector a has 0/1 coordinates with l ones Indeed, without loss of generality assume that a 1 = a = = a l = 1 and a l+1 = a l+ = = a n = 0 Then, n a iε i = l ε i Theorem 1 imlies the following estimate 5

6 Corollary 34 Let a R N, N = n, be a vector with l coordinates equals to one and N l zero coordinates Then, for, N E S a i ε i (l) / Proof of Proosition 31 Denote X := n a iε i = l a iε i Since the vector a has 0/1 coordinates with l ones, a = l For every 0 k l we comute the robability q k that exactly k of ε 1,ε,,ε l equals to one (in that case X = k l) Since S = n ε i = 0, in order to get k ones, we have to choose k ones out ( of )( ε 1,ε,,ε ) l and n k l n l ones out of ε l+1,ε l+,,ε n This gives us choices Since { k n k n ( ) n Ω = ε { 1,1} n ε i = 0} =, we obtain that q k = k, ie n X = (ξ E ξ), where ξ has hyergeometric distribution with arameters (n, n, l) Therefore, Corollary 33 imlies (E ξ Eξ ) 1/ l We would also like to note that Proosition 31 can be roved directly Below we rovide such a direct roof, which gives in lace of in front ( ) / l of This roof is of interest as it can be extended to slightly more 4 general case (see Remark 35) and can be used in another aroach to the main roblem (see Remark 43) Direct roof of the Proosition 31 From Stirling s formula together with the observation that πn ( n n) /4 n increases, we observe that ( ) n n n πn n πn Using this, we obtain ) ( n l n k ( n n) ( n l n l ) ( n n) n l π(n l ) πn 1 (8) n l 6

7 Therefore E ξ Eξ = 1 ( l l k l k k=0 )( n l n k ( n n ) ) l+ l ( ) l k l = k E S l, where S l is a sum of l iid Rademacher random variables By Khinchine inequality (1), we have (E S l ) 1/ l Thus, E ξ Eξ k=0 ( ) / l 4 Remark 35 The above roof can be extended to slightly larger class of ( hyergeometric )/( ) random variables Note that [ ( the)] roof works whenever N l N π N 1 Thus, if l N log n k n, then n E ξ Eξ (l/4) for a (N, n, l) hyergeometric random variable ξ 4 Concluding Remarks In this section we discuss Problem 13 A ossible aroach to this roblem is to use the concentration on the grou Π N (endowed with the distance d N (σ,π) = {i : σ(i) π(i)} ) The following Theorem was roved by Maurey ([11], see also [15]) Theorem 41 Let f : Π N R be 1-Lischitz function Then for all t > 0 µ({σ : f(σ) Ef t}) e t /(3N) (9) Let us mention here, the following oen question, osed by G Schechtman in [15]: Is there an equivalent (with constants indeendent of N) metric on Π N for which the isoerimetric roblem can be solved? Theorem 41 imlies the following estimate 7

8 Corollary 4 Let a,b R N Let f : Π N R be defined by f(σ) := a σ(i) b i (10) Then (E f ) 1/ E f +4 N a b (11) Proof It is easy to see that f is a Lischitz function with Lischitz constant a b, indeed, f(σ) f(π) a σ(i) b i a π(i) b i b i a σ(i) a π(i) a b d N (σ,π) Using Theorem 41 and the bound Γ(x) x x 1 for all x 1 (see for examle [4]), we obtain E f Ef = Thus, 0 µ N ( f Ef t )dt e t /(3N a b ) t 1 dt 0( 4 Γ N ) / a b 4 N / / a b (E f ) 1/ E f +4 N a b E f +4 N a b Remark 43 In the case when b i = ±1 with condition b i = 0, Corollary 4 gives an additional factor N in the uer estimate in (4) Using the chaining argument similar to the one used in [1,, 3] and Proosition 31, the factor N can be reduced to lnn (the details will be rovided in [17]) 8

9 Remark 44 It would be nice to obtain the uer bound in Corollary 4 with constant indeendent of N Acknowledgement: I would like to exress my dee gratitude to my suervisor AE Litvak for encouragement and helful discussions I am also thankful to M Rudelson for his valuable suggestions and very useful discussions References [1] R Adamczak, AE Litvak, A Pajor, N Tomczak-Jaegermann, Quantitative estimates of the convergence of the emirical covariance matrix in Log-concave Ensembles, Journal of AMS, 34 (010), [] R Adamczak, AE Litvak, A Pajor, N Tomczak-Jaegermann, Restricted isometry roerty of matrices with indeendent columns and neighborly olytoes by random samling, Constructive Aroximation, 34 (011), [3] R Adamczak, R Lata la, AE Litvak, A Pajor, N Tomczak- Jaegermann, Tail estimates for norms of sums of log-concave random vectors, Proc London Math Soc, to aear [4] G D Anderson, S L Qiu, A monotoneity roerty of the gamma fuction, Proc AMS, 15 (1997), [5] D Chafai, O Guédon, G Lecué, A Pajor, Interaction between comressed sensing, random matrices and high dimensional geometry, Panoramas et Synthses, 37 (01) [6] D J H Garling, Inequalities: A Journey into Linear Analysis, Cambridge University Press, Cambridge, 007 [7] O Guedon, P Nayar, T Tkocz, Concentration inequalities and geometry of convex bodies, Extended notes of a course, Polish Academy of Sciences of Warsaw, to aear (htt://erso-mathuniv-mlvfr/users/guedonolivier/listeubhtml) 9

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