Delocalisation of one-dimensional marginals of product measures and the capacity of LTI discrete channels
11 Delocalisation of one-dimensionalmarginals of product measures and thecapacity of LTI discrete channels
Maxime Bombar and Alexander Fish
Abstract —We consider discrete linear time invari-ant (LTI) channels satisfying the phase independence(PI) assumption. We show that under the PI assump-tion the capacity of LTI channels is positive. The maintechnical tool that we use to establish the positivityof the capacity is the delocalisation theorem for one-dimensional marginals of the product measure due toBall and Nazarov. We also prove two delocalisationresults that can be seen as extensions of Ball-NazarovTheorem.
Index Terms —LTI channels, PI assumption, Shan-non capacity, delocalisation. I. I NTRODUCTION
In this work we consider linear time-invariant(LTI) discrete channels. Given a sequence x ∈ C N at the transmitter side, we assume that at the re-ceiver we obtain the sequence y ∈ C N given by: y [ n ] = N − (cid:88) m =0 a m x [ n − m mod N ] + ω [ n ] , (1) n = 1 , . . . , N, where { a m } ’s are the attenuation coefficients, and ω is the additive white Gaussian noise (AWGN).The parameter N will be the total length of thesequence. We will see in Section II that in the caseof wireless communication we have N = T W ,where W is the bandwidth and T is the time ofthe modulation. We will assume that the signal to M. Bombar is with ENS Paris-Saclay, Cachan, France, e-mail:[email protected]. Fish is with School of Mathematics and Statistics, Univer-sity of Sydney, Sydney, NSW 2000, Australia, e-mail: alexan-der.fi[email protected]. noise ratio (SNR) is fixed and equal to P . Letus denote by α m = | a m | , and X m = arg ( a m ) , m = 0 , . . . , N − . Then we rewrite equation (1)by use of the operator notation: y = H N ( x ) + ω. (2)The operator H N : C N → C N is given by: H N = N − (cid:88) m =0 α m X m π m , (3)where π m is the circular shift by m on C N , i.e., π m ( x )[ n ] = x [ n − m mod N ] , n = 0 , . . . , N − .We introduce the following assumption for re-quirements on the channel operator. Definition I.1 (PI assumption) . We will say thatthe channel operator H N given by Equation (3)satisfies the phase independence (PI) assumption,if all X m ’s are independent identically distributedrandom variables having uniform distribution on theunit circle. Remark I.2.
In the case of a channel appearingin wireless communication under assumption thatthe carrier frequency f c is much higher than thebandwidth W , it is natural to assume that arg ( a m ) ’sare independent identically distributed random vari-ables having uniform distribution on the unit circle(see Section II).We further assume that at the receiver the channelestimation is performed, and we know the atten-uation coefficients { a m } ’s. We will say that an N -dimensional probability distribution x satisfies a r X i v : . [ c s . I T ] S e p A P N , if E ( tr ( xx t )) = P N . We will denote thelatter by x ∈ A P N . The main result of the paper isthe following.
Theorem I.3.
The asymptotic Shannon capacity ofthe channel given by (2) and (3), and satisfying thePI assumption is positive: lim inf N →∞ N sup x ∈ A PN I ( x ; ( y , H N )) > . Here I ( x , y ) denotes the mutual information of thevectors x and y , see [9].In order to prove Theorem I.3 we use the prob-abilistic result of Ball and Nazarov [3] on uni-form delocalisation of one-dimensional marginalsof product measure for a given delocalised proba-bility distribution in R d . Let us recall that for a R d -valued random variable X , the Levy concentrationfunction of X is defined for every ε > by C ε ( X ) = sup z ∈ R d P ( (cid:107) X − z (cid:107) ≤ ε ) . To state Ball-Nazarov theorem, we will also haveto define the class F of the random variables: F = { N (cid:88) n =1 a n X n | N ≥ , N (cid:88) n =1 | a n | = 1 ,X , . . . , X N are i.i.d. as X } . Then the following theorem holds true:
Theorem I.4 (Ball-Nazarov [3]) . Let X be a R d -valued random variable. Then there exists a univer-sal constant A > such that for every ε > wehave sup S ∈F C ε ( S ) ≤ AC ε ( X ) . Remark I.5.
The quantitative bound A = √ inthe case where X has a bounded density in R has been recently obtained by Rudelson and Ver-shynin (Theorem 1.2 in [8]). Moreoever, Theoremof Rudelson-Vershynin does not require from therandom variables to be identically distributed, butonly independent. Their theorem is a corollary ofK. Ball Theorem, see [2], on the slice of a maximalarea of the unit cube in R N . In this paper, we prove a stronger version ofTheorem I.4 in R under more restrictive condi-tions. We will identify R with the complex plane C . We assume that X is uniformly distributed onthe unit circle in C , and there exists ε > suchthat a projection of the unit weight vector on any coordinates has (cid:96) -norm at most − ε . Moreprecisely, let us denote by S ,N the collection of all -dimensional subspaces in R N generated by any vectors from the standard basis { e , . . . , e N } of R N . We define the class S ,ε = { ( a , . . . , a N ) ∈ S N − | (cid:107) π E ( a ) (cid:107) ≤ − ε , ∀ E ∈ S ,N , N ≥ } , where π E denotes the orthogonal projection on thesubspace E , and S N − is the unit sphere in R N defined by S N − = { ( a , . . . , a N ) ∈ R N | N (cid:88) k =1 a k = 1 } . We think on the set S ,ε as the set of non too sparsepower profiles of the channel operator H N . Wedefine the collection of C -valued random variables F ,ε to be: F ,ε = { a X + . . . + a N X N ∈ F | ( a , . . . , a N ) ∈ S ,ε } . We prove the following result which is slightlybetter than the bound that we can deduce fromTheorem I.4. Theorem I.6.
Let X be a random variable uni-formly distributed on the unit circle in C , and ε > . Then there exists a constant B > suchthat for every ε > we have sup S ∈F ,ε C ε ( S ) ≤ Bε . Theorem I.6 is sharp. We know that the -foldconvolution of the uniform probability measure onthe unit circle has density function f ( x ) in the disk Our method can be also used to provide an alternative proofof Theorem I.4 in the special case, where X is uniformlydistributed on the unit circle in C . of radius in C with singularities on the unit circle.More precisely, there are constants C, c > suchthat for x ∈ C close to the unit circle we have [4]: c | log(1 − | x | ) | ≤ f ( x ) ≤ C | log(1 − | x | ) | . The latter implies that if S = ( X + X + X + X ) , where X , . . . , X are independent uniformlydistributed on the unit circle, then there exists K > such that P ( | S | ≤ ε ) ≥ Kε.
Another consequence of our approach to thedelocalisation is a multi-dimensional analog ofRudelson-Vershynin’s result (Theorem 1.2 in [8]).
Theorem I.7.
Let
K, L, M, δ, η > . Let ( X n ) be R d -valued independent random variables, havingdensities bounded by K , and Γ n be the covariancematrices of X n ’s. Assume that for all n , all theentries of Γ n ’s are bounded by L , det Γ n ≥ δ ,and E [ (cid:107) X n (cid:107) η ] ≤ M . Then there exists C > such that for every N ≥ , the density function of a X + . . . + a N X N is bounded by CK , providedthat (cid:80) Nn =1 a n = 1 .II. L INEAR T IME I NVARIANT (LTI)
DISCRETECHANNELS
In this Section we will justify the channel modelgiven by the Equations (2) and (3) in the scenarioof wireless communication in urban environment.Our assumptions are the following: • The communication is done between a trans-mitter and a receiver by use the frequency band (cid:2) f c − W , f c + W (cid:3) , where f c is the carrierfrequency and W is the bandwidth. Also, weassume f c (cid:29) W . • The environment, the transmitter and the re-ceiver are almost static (have very low speed).As a result the signal experiences a negligibleDoppler effect. The environment has manyobstacles, and there is no line of sight betweenthe transmitter and the receiver. Let µ n := E ( X n ) be the expection of X n . Then Γ n := E (( X n − µ n )( X n − µ n ) t ) is the covariance matrix of X n . • The processing is performed at the baseband (cid:2) − W , W (cid:3) .Let us assume that at the transmitter side wehave a sequence S [ n ] = b n , n = 1 , , . . . , of real (complex) numbers that we would like tosend to a receiver. It is a standard procedure inwireless communication, to use some Digital-to-Analog transform, in order to generate a function s ( t ) with a spectral profile in a given band whichencompasses the information. We will stick in thispaper to the Shannon-Nyquist transform given by s ( t ) = exp ( if c t ) (cid:32)(cid:88) m b m sinc ( − W t + m ) (cid:33) , where sinc ( t ) = sin πtπt . It follows from the proper-ties of the sinc function that: • The signal s ( t ) satisfies that (cid:98) s ⊂ (cid:2) f c − W , f c + W (cid:3) . • The Inverse (Analog-to-Digital) operator isgiven by the sampling s ( t ) at the time slots { W , W , . . . } . In other words, we have s (cid:16) nW (cid:17) = b n = S [ n ] , for n ≥ . We use the model of multi-path propagation of asignal [12]. Assume that at the transmitter side wegenerate the signal s ( t ) as above. Then for any path L of length (cid:96) we obtain at the receiver the signal: r L ( t ) = α L s ( t − (cid:96)/c ) + ω L ( t ) , where c denotes the speed of light, α L ∈ R is theattenuation coefficient which depends on the pathlength (decays as (cid:96) ) and on the environment, and ω L denotes the additive noise along the path L .We assume that the received signal r L ( t ) isshifted to the baseband first, and then, we sample exp ( − if c t ) r L ( t ) at the time slots { W , W , . . . } . Wealso assume, as in [6], that the time shift (cid:96)c lies onthe lattice W Z , i.e., there exists n L ∈ Z such that (cid:96)/c = n L W . Then, at the receiver side, we obtain thefollowing sequence: R L [ n ] = exp ( − if c n L /W ) r L (cid:16) nW (cid:17) = exp ( − if c n L /W ) α L b n − n L + ω L ( n/W ) ,n = 1 , , . . . Therefore, we have R L [ n ] = exp ( − if c n L /W ) α L S [ n − n L ]+ ω L ( n/W ) . By our assumptions, f c (cid:29) W , and therefore,the phase f c n L /W will change drastically if wereplace n L by another close to it integer. So, weapproximate R L by the following: R L [ n ] ≈ α L X L S [ n − n L ] + ω L ( n/W ) , where X L is uniformly distributed random variableon the unit circle in C .Finally, taking all the paths from the transmitterto the receiver, by the law of superposition, weobtain the sequence R given by: R [ n ] = (cid:88) k ∈ N α k X k S [ n − k ] + ω [ n ] , n = 1 , , . . . where ω [ n ] = (cid:80) L ω L ( n/W ) . We will assumethat the phases X k are independent uniformly dis-tributed on the unit circle in C , since they belongto different paths, and therefore are not related oneto each other. It is a standard assumption that all ω [ n ] ’s are independent complex Gaussian randomvariables with zero mean and variance one.By use of the trick of periodic prefix, like in [6],using a transmission of a finite duration time T ,we can assume that the transmitted and receivedsequences, S and R respectively, have length N = T W , and the relation between S and R is describedby equation: R [ n ] = N − (cid:88) k =0 α k X k S [ n − k mod N ] + ω [ n ] , (4)where n = 0 , , . . . , N − . Notice, that the Equation(4) coincides with the model Equation (3) of thechannel given in Section I. III. O N THE CAPACITY OF
LTI
CHANNELSUNDER THE PI ASSUMPTION
For each N , let H N be the channel operatordefined by Equation (3). It is a standard fact that H N is diagonalisable and its eigenvectors are theexponential functions e ( N ) k ∈ C N given by e ( N ) k [ m ] = e πikmN , m ∈ { , , . . . , N − } . Therefore, the matrix of the operator H N in thebasis of the exponentials is λ ( N )1 . . . . . . . . . λ ( N ) N where the λ ( N ) k are the eigenvalues of H N givenby: λ ( N ) k = N − (cid:88) m =0 α m X m e − πikmN , k = 1 , , . . . , N. (5) Remark III.1.
Since the uniform distribution onthe circle is invariant by rotations, ( λ ( N ) k ) is asequence of identically distributed random variables(but not independent) for each N .We express ω := ω e + · · · + ω N e N , where e , . . . , e N is the basis of the exponentials. Since ω is a white Gaussian noise of total power N , the ω k are distributed according to the complex symmetricGaussian distribution, with 0 mean and variance .We assume that the noise affects each coordinateindependently, ie E [ ωω t ] = I N . A. Proof of Theorem I.3
We denote by C N = sup x ∈ A PN I ( x ; ( y , H N )) . Inthis Section we prove the lower bound for C N which implies Theorem I.3.In [11], E. Telatar studied AWGN linear chan-nels with random attenuation coefficient matrixand proved that if x is constrained to have covari-ance matrix Q then the choice of x that maximises the noise is assumed to be an additive white Gaussian noiseand the operator acts linearly on the signals I ( x ; ( y , H N )) , where H N is random and known atthe receiver, is the circularly symmetric complexGaussian of covariance Q and that sup x ∈ A PN I ( x ; ( y , H N ))= sup Q E [log det( I N + H N QH † N )] , where the supremum is taken over the choices ofnon-negative definite Q subject to tr ( Q ) ≤ P N ,and H † N denotes the adjoint operator to H N .Since for any matrices A, B with suitable sizes det( I + AB ) = det( I + BA ) , we can rewrite C N as follows: C N = sup Q E (cid:104) log det( I N + QH † N H N ) (cid:105) = sup Q E (cid:2) log det( I N + Q | H N | ) (cid:3) , (6)where | H N | denotes the diagonal matrix whosecoefficients are | λ i | ’s. Now, taking Q = P I N wehave: C N ≥ E (cid:2) log det( I N + P | H N | ) (cid:3) = N (cid:88) m =1 E (cid:2) log(1 + P | λ m | ) (cid:3) = N E (cid:2) log(1 + P | λ | ) (cid:3) . In the last transition we used Remark III.1.Then for any ε > we have: log(1 + P | λ | ) ≥ log(1 + P | λ | ) {| λ |≥ ε } ≥ log(1 + P ε ) {| λ |≥ ε } The linearity and monotonicity of the expectationimply: E (cid:2) log(1 + P | λ | ) (cid:3) ≥ log(1 + P ε ) P {| λ | ≥ ε } . By Theorem I.4, there exists B > such thatfor all ε > we have: P {| λ | < ε } ≤ Bε.
Finally, taking ε = 12 B we obtain: N C N ≥
12 log (cid:18) P B (cid:19) > . Taking the lim inf of the left hand side as N → ∞ concludes the proof of Theorem I.3.IV. O N DELOCALISATION OFONE - DIMENSIONAL MARGINALS OF THEPRODUCT MEASURE A. Preliminaries
The uniform measure on a circle is not abso-lutely continuous with respect to Lebesgue measure.However, we can compute the 2-fold convolution ofthis measure to see that it is absolutely continuous(see [4]). By Lemma IV.1, a linear combination ofrandom variables uniformly distributed on the unitcircle also has a density. In Appendix A we willprove the following lemmata:
Lemma IV.1.
Let
X, Y be R d -valued independentrandom variables, such that Y has density. Thenfor every real number a , the random variable aX + Y has density. If, moreover, the density of Y isbounded by M , then the density of aX + Y is alsobounded by M . Remark IV.2.
If both X and Y have a density, thenthis result directly follows from Young inequality. Lemma IV.3.
Let a , . . . , a ∈ C , b , . . . , b > such that for every k, | a k | ≥ b k . Let X , . . . , X beindependent random variables uniformly distributedon the unit circle. Then there exists a function ϕ At this point of the proof, we could, alternatively, useTheorem I.6 instead of Ball-Nazarov Theorem I.4. Our approachwould require the delocalisation estimates for , , and -foldconvolutions of the uniform measure on the unit circle. Indeed, if X is uniformly distributed on the unit circle in C , then there exists B > such that for every ε > we have C ε ( X ) ≤ Bε . which only depends on the b i ’s such that the density f of a X + · · · + a X satisfies: (cid:107) f (cid:107) ∞ ≤ ϕ ( b , . . . , b ) . Lemma IV.4.
Let X be an R d − valued randomvariable having density f , a ∈ R \ { } , b ∈ R d .Then aX + b has density at x ∈ R d equal to | a | d f (cid:18) x − ba (cid:19) .In our work, we use a local R d − valued CentralLimit Theorem that follows from Theorem in [10],see Appendix B. Theorem IV.5 (Local R d − valued CLT) . For each n , let X n,m , ≤ m ≤ n be a triangular array of in-dependent R d − valued, centered, random variableswith covariance matrices C n,m := E [ X n,m X tn,m ] .Let σ n,m := tr ( C n,m ) = E [ (cid:107) X n,m (cid:107) ] .If i ) There exists a symmetric, positive-definite d × d matrix C such that: n (cid:88) m =1 C n,m → C ii ) For any ε > , for any θ ∈ R d , n (cid:88) m =1 E (cid:2) |(cid:104) θ, X n,m (cid:105)| {|(cid:104) θ,X n,m (cid:105)| >ε } (cid:3) → then, the sum S n := X n, + . . . + X n,n convergesin distribution towards the d − variate Gaussian with mean and covariance C .Moreover, if one of the following conditions holdstrue: ) Starting from some n , S n has a density,and there exists an integer p > such that (cid:107) φ n,m (cid:107) pd L p σ n,m is uniformly bounded, where φ n,m denotes the Fourier transform of X n,m . ) All the X n,m ’s have uniformly bounded den-sity.then (cid:107) f n − g C (cid:107) ∞ → , where f n and g C denote the densities of S n and ofthe d − variate Gaussian with mean and covariancematrix C , respectively. B. Proof of Theorem I.6
Assume that, contrary to the assertion of Theo-rem I.6, for each n there exist z ∈ C , ( a ( n ) ) ∈ S ( n − , ε n → such that P {| S n − z | < ε n } ε n → + ∞ , where S n = (cid:80) nk =1 a ( n ) k X k . In the following, f n will denote the density function of S n (for n ≥ )and g σ will be the density function of a circularsymmetric Gaussian with 0 mean and variance σ . Case 1: Uniform decay of the coefficients : If we have max ≤ k ≤ n | a ( n ) k | → as n → ∞ , (7)then by Theorem IV.5 (see Appendix C), (cid:107) f n − g (cid:107) ∞ → . Therefore, (cid:107) f n (cid:107) ∞ ≤ + (cid:107) g (cid:107) ∞ for n sufficientlylarge. Then P {| S n − z | < ε n } = (cid:90) B ( z,ε n ) f n ( x ) dx ≤ M ε n for some constant M independent of n which leadsto a contradiction. Case 2: Up to bounded away from zerocoefficients : There exist k ∈ { , . . . , } and δ , . . . , δ k > ,and there exist a sequence ϕ ( n ) and i ( n ) < · · · such that for j ∈ { , . . . , } , | a ( n ) i j | ≥ δ j for infinitely many n . Without lossof generality, we can assume that i j = j andthe inequalities hold for all n . Then, by LemmaIV.3, the density of a ( n )1 X + · · · + a ( n )5 X isbounded by ϕ ( δ , . . . , δ ) . Hence, by Lemma IV.1, S n has density bounded by ϕ ( δ , . . . , δ ) for all n . This leads to a contradiction, and completes theproof. C. Proof of Theorem I.7
The approach is similar to the proof of The-orem I.6. Assume that contrary to the assertionof Theorem I.7, there exists a sequence of S n = (cid:80) nk =1 a ( n ) k X k whose density functions f n are un-bounded. By Lemma IV.4 we may assume that µ k = E ( X k ) = 0 for all k .In the following we denote by D δ the set ofall symmetric, positive-definite d × d matrices withdeterminant at least δ . It is a closed and convex set(see proof of Theorem 1 in [7]).Let C n = n (cid:88) k =1 | a ( n ) k | Γ k be the covariance matrix of S n . By convexity of D δ we have C n ∈ D δ for every n . Moreover, all entries of C n are bounded by L . Bythe compactness argument we can assume withoutloss of generality that: C n → C ∈ D δ . In particular, the matrix C is positive-definite. Uniform decay of the coefficients : Let’s assume that max ≤ k ≤ n | a ( n ) k | → as n → ∞ . (8)Then, by Theorem IV.5 (see Appendix C) (cid:107) f n − g C (cid:107) ∞ → . Therefore, we have (cid:107) f n (cid:107) ∞ ≤ + (cid:107) g C (cid:107) ∞ for n sufficiently large which leads to a contradiction. At least bounded away from 0 coefficient : If condition (8) is not true, then there exist γ > ,and sequences ϕ ( n ) and i ( n ) such that | a ( ϕ ( n )) i ( n ) | ≥ γ Without loss of generality, we can assume that i ( n ) = 1 for every n . It follows from Lemma IV.1and IV.4 that for every n the density of S ϕ ( n ) isbounded by γ − d K which leads to a contradiction.V. C ONCLUDING R EMARKS
The main difference between our work andknown results for LTI discrete channels is in thefollowing: • We argue in Section II that it is reasonable,especially, in the scenario of wireless com-munication in urban environment, to assumethat the channel operator H N satisfies the PIassumption. We have no assumptions on thepower profile of the channel operator, i.e., noassumptions on the vector of ( α m ) ’s. • The exact variational formula for the capacityis well known in the literature [1], [11], [13].See also Formula (6) in Section III, and For-mula 1.4 in [13]. We provide a positive lower bound on the capacity under minor assump-tions on the channel operators H N . • We relate the information theoretical problemat the hand to well established Delocalisationproblem in Probability, and provide a new res-olution of the latter through Lindeberg-FellerLocal Central Limit Theorem IV.5. We alsoestablish a completely new multi-dimensionaldelocalisation result, see Theorem I.7, whichmight have further applications in InformationTheory. A
PPENDIX AP ROOF OF L EMMATA
IV.1
AND
IV.3
Proof (Lemma IV.1) . • Absolute continuity of aX + Y . First we needto show that P aX + Y is absolutely continuous.Let f denote the density of Y and let A be aBorel set with Lebesgue measure 0. Then byindependence we have: P aX + Y ( A ) = (cid:90) R d P ( Y ∈ A − x ) d P aX ( x )= (cid:90) R d (cid:90) A − x f ( y ) dyd P aX ( x )= 0 In the last transition we used the fact that theLebesgue measure is invariant by translation,so A − x has Lebesgue measure for all x ∈ R d . • Bound on the density of aX + Y . Again,using the independence of X and Y and Fubinitheorem, the following holds for any Borel set A : P aX + Y ( A ) = (cid:90) R d P ( Y ∈ A − x ) d P aX ( x )= (cid:90) R d (cid:90) A − x f ( y ) dyd P aX ( x )= (cid:90) R d (cid:90) A f ( u − x ) dud P aX ( x )= (cid:90) A (cid:90) R d f ( u − x ) d P aX ( x ) du By uniqueness of the Radon-Nikodym deriva-tive for almost every u ∈ R d , the density ofthe sum is defined by g ( u ) := (cid:90) R d f ( u − x ) d P aX ( x )= E ( f ( u − aX )) Hence we have (cid:107) g (cid:107) ∞ ≤ E ( M ) = M. To prove Lemma IV.3 we will need some esti-mates on Bessel function of the first kind. Denoteby µ r the uniform measure on the circle centeredat of radius r , and by (cid:99) µ r its Fourier transformdefined for all x ∈ R as (cid:99) µ r ( x ) = (cid:90) rS e − i (cid:104) y,x (cid:105) dµ r ( y ) Lemma A.1. (cid:99) µ r is radial and for all x ∈ R : (cid:99) µ r ( x ) = 12 π (cid:90) π e − ir | x | cos ( θ ) dθ The latter is also known as J ( r | x | ) where J is aBessel function of the first kind. Proof (Lemma A.1) . It follows directly by passingto polar coordinates: (cid:99) µ r ( x ) = 12 π (cid:90) π e − ir | x | cos ( θ − Arg ( x )) dθ = 12 π (cid:90) π e − ir | x | cos ( θ ) dθ, where the last transition is made by rotation invari-ance of the uniform measure on the unit circle. Lemma A.2.
Let X be an R d − valued randomvariable and let ϕ denote its Fourier transformdefined as ϕ ( y ) := E ( e − i (cid:104) y,X (cid:105) ) . If ϕ ∈ L ( R d ) then X has a density, bounded by (2 π ) − d (cid:107) ϕ (cid:107) L . Proof (Lemma A.2) . This is a standard fact about R d -valued random variables. Lemma A.3.
There exists K such that for all r > , | J ( r ) | ≤ min (cid:16) , K √ r (cid:17) . Proof (Lemma A.3) . J ( r ) = 12 π (cid:90) π/ − π/ e − ircos ( θ ) dθ + 12 π (cid:90) π/ π/ e − ircos ( θ ) dθ = 12 π ( I + I ) By the Principle of Stationary Phase the followingestimate holds as r → ∞ , I k = (cid:114) πr cos (cid:16) r − π (cid:17) + O (cid:18) r (cid:19) Hence J ( r ) = (cid:114) πr cos (cid:0) r − π (cid:1) + O (cid:0) r (cid:1) . There-fore, √ r | J ( r ) | is bounded. We can take K :=sup r> √ r | J ( r ) | . Lemma A.4. (cid:99) µ a ∈ L ( R ) for any a > . Proof (Lemma A.4) . (cid:99) µ a is radial, continuous on R and by Lemma A.3, | (cid:99) µ a ( x ) | = O (cid:18) | x | / (cid:19) as | x | → ∞ , which is integrable on R . Therefore, sois | (cid:99) µ a | .We are now able to prove Lemma IV.3: Proof (Lemma IV.3) . Let µ a denote the probabilitymeasure of aX . By independence, the probabilitymeasure µ of the sum a X + . . . + a X is theconvolution µ a ∗ . . . ∗ µ a . Then, (cid:98) µ = (cid:100) µ a . . . (cid:100) µ a .It follows from Lemmata A.1 and A.3 that | (cid:98) µ ( x ) | = | J ( | a x | ) . . . J ( | a x | ) |≤ (cid:89) k =1 min (cid:32) , K (cid:112) a k | x | (cid:33) ≤ (cid:89) k =1 min (cid:32) , K (cid:112) b k | x | (cid:33) Hence, (cid:107) (cid:98) µ (cid:107) L ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:89) k =1 min (cid:32) , K (cid:112) b k | x | (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L Therefore, the result follows from Lemma A.2. A
PPENDIX BP ROOF OF T HEOREM
IV.5In order to prove Theorem IV.5, it will suffice toprove that ( S n ) converges in distribution towardsthe Gaussian. This will hold true using an R d version of Lindeberg-Feller Theorem that we recallhereafter. We will denote by = ⇒ the convergencein distribution. Theorem B.1 (Lindeberg-Feller CLT) . For each n let X n,m , ≤ m ≤ n be independent real-valuedrandom variables with E [ X n,m ] = 0 . If1) n (cid:88) m =1 E [ X n,m ] → σ as n → ∞
2) For each ε > , n (cid:88) m =1 E (cid:104) | X n,m | {| X n,m | >ε } (cid:105) → , then S n := X n, + · · · + X n,n = ⇒ N (0 , σ ) . Proof (Theorem B.1) . The proof can be found inSection 2.4 of [5].Using Cram´er-Wold’s characterisation of conver-gence in distribution in R d , we can easily generaliseTheorem B.1 to random vectors. Theorem B.2 (Cram´er-Wold) . Let ( X n ) be a se-quence of random vectors in R d . We have X n = ⇒ X ∞ if and only if (cid:104) θ, X n (cid:105) = ⇒ (cid:104) θ, X ∞ (cid:105) for all θ ∈ R d . Theorem B.3 ( R d -valued Lindeberg-Feller CLT) . For each n let X n,m , ≤ m ≤ n, be independent R d -valued random vectors with E [ X n,m ] = 0 . Let C n,m := E [ X n,m X tn,m ] be the covariance matrixof X n,m .If we have i ) There exists C a symmetric, positive-definite, d × d matrix such that : n (cid:88) m =1 C n,m → C ii ) For any ε > , for any θ ∈ R d , n (cid:88) m =1 E (cid:2) |(cid:104) θ, X n,m (cid:105)| {|(cid:104) θ,X n,m (cid:105)| >ε } (cid:3) → then S n := X n, + . . . + X n,n converges in distribu-tion towards a Gaussian with mean and covariancematrix C .Theorem IV.5 is now a particular case of Theo-rem of [10]. A PPENDIX
CIn what follows, we will check that all therequirements of Theorem IV.5 are fulfilled.
A. In the proof of Theorem I.6:
For the uniform distribution on the unit circle,the real and imaginary parts are uncorrelated. More-over, their first moment is and their variance areequal. In what follows, let us denote by X n,m = a ( n ) m X m , where X n ’s are independent uniformlydistributed random variables on the unit circle in C .For each m, and n we have C n,m = E ( X n,m X tn,m ) = 12 (cid:12)(cid:12)(cid:12) a ( n ) m (cid:12)(cid:12)(cid:12) I , and therefore σ n,m = tr ( C n,m ) = (cid:12)(cid:12)(cid:12) a ( n ) m (cid:12)(cid:12)(cid:12) . Hence, n (cid:88) m =1 C n,m = 12 I . (9)If we have max ≤ m ≤ n (cid:12)(cid:12)(cid:12) a ( n ) m (cid:12)(cid:12)(cid:12) → , as n → ∞ , then for every ε > the event (cid:110)(cid:12)(cid:12)(cid:12) a ( n ) m X m (cid:12)(cid:12)(cid:12) > ε (cid:111) is empty starting from some n . Then, Lindeberg-Feller’s condition ii ) in Theorem IV.5 holds true.Moreover, by the results of Appendix A, theassumption on the Fourier transforms is satisfiedfor p = 5 , so the requirements of Theorem IV.5hold true. B. In the proof of Theorem I.7:
Let θ ∈ R d and ε > . Then we have thefollowing: |(cid:104) θ, a ( n ) k X k (cid:105)| {|(cid:104) θ,a ( n ) k X k (cid:105)| >ε } = | a ( n ) k | |(cid:104) θ, X k (cid:105)| (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ( n ) k (cid:104) θ, X k (cid:105) ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η > ≤ ε η (cid:12)(cid:12)(cid:12) a ( n ) k (cid:12)(cid:12)(cid:12) η |(cid:104) θ, X k (cid:105)| η . Denote by γ n = max ≤ k ≤ n | a ( n ) k | and assume that γ n → as n → ∞ . Then, by taking the expectationand using Cauchy-Schwarz inequality we have: n (cid:88) k =1 E [ |(cid:104) θ, a ( n ) k X k (cid:105)| |(cid:104) θ,a ( n ) k X k (cid:105)| >ε ] ≤ ε η n (cid:88) k =1 (cid:12)(cid:12)(cid:12) a ( n ) k (cid:12)(cid:12)(cid:12) η E [ |(cid:104) θ, X k (cid:105)| η ] ≤ (cid:107) θ (cid:107) η ε η γ ηn n (cid:88) k =1 (cid:12)(cid:12)(cid:12) a ( n ) k (cid:12)(cid:12)(cid:12) E (cid:2) (cid:107) X k (cid:107) η (cid:3) ≤ M (cid:107) θ (cid:107) η ε η γ ηn → as n → ∞ Recall, we derived in the Proof of Theorem I.7 thatthe covariance matrices of the sums S n convergeto a positive-definite matrix C . Therefore, all therequirements of Theorem IV.5 are satisfied.A CKNOWLEDGMENT
The authors would like to thank Ben Goldys,Georg Gottwald, Uri Keich, Mark Rudelson andOfer Zeitouni for sharing their ideas on the varioussubjects related to the work. The first named authorwould like to thank the hospitality of the Universityof Sydney where this work has been carried out.R
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Maxime Bombar is a 3rd year student at ´Ecole NormaleSup´erieure Paris-Saclay, France, studying mathematics and com-puter science.