Sums of independent random variables
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1 3 Sums of indeendent random variables This lecture collects a number of estimates for sums of indeendent random variables with values in a Banach sace E. We concentrate on sums of the form N γ nx n, where the γ n are real-valued Gaussian variables and the x n are vectors in E. As we shall see later on such sums are the building blocks of general E-valued Gaussian random variables and, erhas more imortantly, stochastic integrals of E-valued ste functions are of this form. Furthermore, they are used in the definition of various geometric roerties of Banach saces, such as tye and cotye. The highlights of this lecture are the Kahane contraction rincile Theorem 3.1), a covariance domination rincile Theorem 3.9) and the Kahane- Khintchine inequalities Theorems 3.11 and 3.1). 3.1 Gaussian sums We begin with an imortant inequality for sums of indeendent symmetric random variables, due to Kahane. Theorem 3.1 Kahane contraction rincile). Let X n ) be a sequence of indeendent symmetric E-valued random variables. Then for all a 1,..., a N R and 1 <, a n X n max a n ) E X n. 1 n N Proof. For all ε 1,..., ε N ) { 1, +1} N the E N -valued random variables X 1,..., X N ) and ε 1 X 1,...,ε N X N ) are identically distributed and therefore ε n X n = E X n.
2 3 3 Sums of indeendent random variables For the general case we may assume that a n 1 for all n = 1,...,N. Then a = a 1,...,a N ) is a convex combination of the N elements of { 1, +1} N, say a = N j=1 λj) ε j). Hence, N a n X n = E j=1 λ j) N N E λ j) N E = N j=1 j=1 N j=1 λ j) N ε j) n X n ε j) n X n ) ε j) n X n λ j) X n = E X n, where the third ste follows from the convexity of the function t t or an alication of Jensen s inequality). As an alication of the Kahane contraction rincile we shall rove an inequality which shows that Rademacher sums have the smallest L -norms among all random sums. Rademacher sums are easier to handle than the Gaussian sums in which we are ultimately interested, and, as we shall see, there are various techniques to ass on results for Rademacher sums to Gaussian sums. Let us begin with a definition. An { 1, +1}-valued random variable r is called a Rademacher variable if P{r = 1} = P{r = +1} = 1. Throughout these lectures, the notation r n ) will be used for a Rademacher sequence, that is, a sequence of indeendent Rademacher variables. Theorem 3. Comarison). Let ϕ n ) be a sequence of indeendent symmetric integrable real-valued random variables satisfying E ϕ n 1 for all n 1. Then for all x 1,...,x N E and 1 < we have r n x n E ϕ n x n. The roof of this theorem relies on an auxiliary lemma, for which we need two definitions based on the following easy observation: if X 1,..., X N are random variables with values in E 1,...,E N, then X 1,...,X N ) is a random variable with values in E 1 E N.
3 3.1 Gaussian sums 31 Definition 3.3. Two families of random variables X i ) i I and Y i ) i I, where I is some index set and X i and Y i take values in a Banach sace E i, are identically distributed if for all choices of i 1,..., i N I the random variables X i1,..., X in ) and Y i1,..., Y in ) are identically distributed. Note that by Proosition.13, if X i ) i I and Y i ) i I are families of indeendent random variables such that X i and Y i are identically distributed for all i I, then X i ) i I and Y i ) i I are identically distributed. Definition 3.4. Two families of random variables X i ) i I and Y j ) j J, where I and J are index sets, X i takes values in E i for all i I and Y j takes values in F j for all j J, are indeendent of each other if for all choices i 1,..., i M I and j 1,..., j N I the random variables X i1,..., X im ) and Y j1,...,y in ) are indeendent. Lemma 3.5. Let ϕ n ) be a sequence of indeendent symmetric real-valued random variables and let r n ) be a Rademacher sequence indeendent of ϕ n ). The sequences ϕ n ) and r n ϕ n ) are identically distributed. Proof. By indeendence and symmetry we have P{r n ϕ n B} = P{r n = 1, ϕ n, ϕ n B} + P{r n = 1, ϕ n <, ϕ n B} + P{r n = 1, ϕ n, ϕ n B} + P{r n = 1, ϕ n <, ϕ n B} = 1 P{ϕ n, ϕ n B} + 1 P{ϕ n <, ϕ n B} + 1 P{ϕ n, ϕ n B} + 1 P{ϕ n <, ϕ n B} = 1 P{ϕ n, ϕ n B} + 1 P{ϕ n >, ϕ n B} + 1 P{ϕ n, ϕ n B} + 1 P{ϕ n <, ϕ n B} = P{ϕ n B}. Since ϕ n ) and r n ϕ n ) are sequences of indeendent random variables, the lemma now follows from the observation receding Definition 3.4. Proof Proof of Theorem 3.). We may assume that the sequences ϕ n ) and r n ) are defined on distinct robability saces Ω ϕ and Ω r. By considering the ϕ n and r n as random variables on the robability sace Ω ϕ Ω r, we may assume that ϕ n ) and r n) are indeendent of each other. Since E ϕ ϕ n 1, with the Kahane contraction rincile and Jensen s inequality we obtain Eϕ E r r n x n Er r n ϕ n x n E r E ϕ r n ϕ n x n = Eϕ ϕ n x n,
4 3 3 Sums of indeendent random variables where the last identity follows from Lemma 3.5. A real-valued random variable γ is called standard Gaussian if its distribution has density f γ t) = 1 π ex 1 t ) with resect to the Lebesgue measure on R. For later reference we note that γ is standard Gaussian if and only if its Fourier transform is given by Eex iξγ) = ex 1 ξ ), ξ R. 3.1) The only if statement follows from the identity 1 π ex iξt 1 t )dt = ex 1 ξ ) which can be roved by comleting the squares in the exonential and then shifting the ath of integration from iξ+r to R by using Cauchy s formula; the if art then follows from the injectivity of the Fourier transform Theorem.8). For a standard Gaussian random variable γ we have E γ = 1 π t ex 1 t )dt = t ex 1 π t )dt = /π. 3.) From this oint on, γ n ) will always denote a Gaussian sequence, that is, a sequence of indeendent standard Gaussian variables. From 3.) and Theorem 3. we obtain the following comarison result. Corollary 3.6. For all x 1,...,x N E and 1 <, r n x n π/) E γ n x n. 3.3) The geometric notions of tye and cotye will be introduced in the exercises. Without roof we state the following imortant converse to Corollary 3.6 for Banach saces with finite cotye. Examles of saces with finite cotye are Hilbert saces, L -saces for 1 <, and the UMD saces which will be introduced in later lectures. Theorem 3.7. If E has finite cotye, there exists a constant C such that for all x 1,..., x N E, γ n x n C r n x n.
5 3.1 Gaussian sums 33 The Kahane-Khintichine inequalities Theorems 3.11 and 3.1 below) can be used to extend this inequality to arbitrary exonents 1 <. The roof of Theorem 3.7 is beyond the scoe of these lectures; we refer to the Notes at the end of the lecture for references to the literature. When taken together, Corollary 3.6 and Theorem 3.7 show that in saces with finite cotye, Gaussian sequences and Rademacher sums can be used interchangeably. Without any assumtions on E, Theorem 3.7 fails. This is shown by the next examle. Examle 3.8. Let E = c and let u n ) be the standard unit basis of c. Then ) c r n u n = E max r n = 1. 1 n N Next we estimate E N γ nu n c from below. First, if γ is standard Gaussian, the inequality 1 x e x imlies P { max 1 n N γ n r } = [ 1 P{ γ > r} ] N ex NP{ γ > r}). For r = 1 log N we estimate P{ γ > 1 log N log N} e 1 x dx π log N 1 π 1 log N e 1 log N = Hence, using the integration by arts formula of Exercise.1, c γ n u n = E = 1 max γ n 1 n N ) P { max 1 n N γ n > r } dr 1 log N log N πn. [ 1 ex NP{ γ > r}) ] dr [ N log N log N 1 ex π 1 log N as N. Similar estimates show that the bound O log N) for N is of the correct order. We conclude this section with an imortant comarison result for Gaussian sums. )]
6 34 3 Sums of indeendent random variables Theorem 3.9 Covariance domination). Let γ m ) and γ n ) be Gaussian sequences on robability saces Ω and Ω, resectively, and let x 1,...,x M and y 1,..., y N be elements of E satisfying M x m, x Then, for all 1 <, y n, x x E. M γ m x m E γ n y n. Proof. Denote by F the linear san of {x 1,...,x M, y 1,..., y N } in E. Define Q L F, F) by Qz := y n, z y n M x m, z x m, z F. The assumtion of the theorem imlies that Qz, z for all z F, and it is clear that Qz1, z = Qz, z 1 for all z 1, z F. Since F is finitedimensional, by linear algebra we can find a sequence x j ) M+k j=m+1 in F such that Q is reresented as Qz = M+k j=m+1 x j, z x j, z F. We leave the verification of this statement as an exercise for the moment and shall return to this issue from a more general oint of view in the next lecture. Now, M+k x m, z = y n, z, z F. 3.4) It follows from 3.1) that the random variables X := M+k γ mx m and Y := N γ n y n have Fourier transforms Eex i X, x ) = = M+k M+k Eex iγ m x m, x ) ex 1 x m, x ) = ex 1 M+k x m, x ) and similarly E ex i Y, x ) = ex 1 N y n, x ). Hence by 3.4) and Theorem.8, X and Y are identically distributed. Thus, for all 1 <,
7 3. The Kahane-Khintchine inequality 35 By Proosition.16, M+k M and the roof is comlete. γ m x m = E γ ny n. M+k γ m x m E γ m x m, 3. The Kahane-Khintchine inequality The main result of this section states that all L -norms of an E-valued Gaussian sum are comarable, with universal constants deending only on. First we rove the analogous result for Rademacher sums; then we use the central limit theorem to ass it on to Gaussian sums. The starting oint is the following inequality, which is a consequence of Lévy s inequality. Lemma 3.1. For all x 1,..., x N E and r > we have { N } [ { N }]. P r n x n > r 4 P r n x n > r Proof. Let us write S n := n j=1 r jx j. As in the roof of Lemma.18 we ut A n := { S 1 r,..., S n 1 r, S n > r}. If for an ω A n we have S N ω) > r, then S N ω) S n 1 ω) > r. Now the crucial observation is that r 1,..., r N ) and r 1,..., r n, r n r n+1,..., r n r N ) are identically distributed; we leave the easy roof as an exercise. From this and the fact that r n = 1 almost surely we obtain PA n { S N S n 1 > r}) = P A n = P A n { N }) r j x j > r j=n { N rn }) r j x j > r j=n { xn = P A n + { xn = P A n + j=n+1 j=n+1 r n r j x j > r }) r j x j > r }) = PA n { x n + S N S n ) > r}),
8 36 3 Sums of indeendent random variables and similarly P{ S N S n 1 > r} = P{ x n + S N S n ) > r}. Hence, by the indeendence of A n and S N S n, PA n { S N > r}) PA n { S N S n 1 > r}) = PA n )P{ x n + S N S n ) > r} = PA n )P{ S N S n 1 > r} PA n )P{ S N > r}, where the last ste follows from Lévy s inequality after changing the order of summation. Summing over n = 1,..., N and using Lévy s inequality once more we obtain P{ S N > r} = PA n { S N > r}) PA n )P{ S N > r} { } = P max S n > r P{ S N > r} 4[P{ S N > r}]. 1 n N We are now ready to rove the following result, which is the Banach sace generalisation due to Kahane of a classical result for scalar random variables of Khintchine. Theorem 3.11 Kahane-Khintchine inequality - Rademacher sums). For all 1, q < there exists a constant K,q, deending only on and q, such that for all finite sequences x 1,..., x N E we have 1 ) r n x n K,q E r n x n q) 1 q. Proof. By Hölder s inequality it suffices to consider the case > 1 and q = 1. Fix vectors x 1,..., x N E. Writing X n = r n x n and S N = N X n, we may assume that E S N = 1. Let j 1 be the unique integer such that j 1 < j. By successive alications of Lemma 3.1 for r > we have P{ S N > j r} 4 j 1 P{ S N > r}) j. Chebyshev s inequality gives rp{ S N > r} E S N = 1. Hence, E S N = t 1 P{ S N > t} dt = j r 1 P{ S N > j r} dr j 4 j 1 r 1 P{ S N > r}) j dr ) 4 1 r 1 P{ S N > r}) dr ) 4 1 P{ S N > r} dr ) 4 1.
9 3. The Kahane-Khintchine inequality 37 The best ossible constants K,q in this inequality are called the Kahane- Khintchine constants. Note that K,q = 1 if q by Hölder s inequality. The bound on K,1 roduced in the above roof is not the best ossible: for instance it is known that K,1 = 1 1 ; see the Notes at the end of the lecture. By an alication of the central limit theorem, the Kahane-Khintchine inequality extends to Gaussian sums: Theorem 3.1 Kahane-Khintchine inequality - Gaussian sums). For all 1, q < and all finite sequences x 1,...,x N E we have 1 ) γ n x n K,q E where K,q is the Kahane-Khintchine constant. γ n x n q) 1 q, Proof. Fix k = 1,,... and define ϕ k) n := 1 k k j=1 r nk+j. For each k we have 1 ϕ k) ) n x n = K,q E k j=1 k j=1 x 1 n ) r nk+j k x 1 n q) q r nk+j k = K,q E ϕ k) n x n q ) 1 q. The roof is comleted by assing to the limit k and using the central limit theorem. The attentive reader has noticed that we are cheating a bit in the above roof, as the usual formulation of the central limit theorem only asserts that lim k ϕ k) 1,...,ϕk) N ) = γ 1,..., γ N ) in distribution, that is, lim k Efϕk) 1,..., ϕk) N ) = Efγ 1,..., γ N ) for all bounded continuous functions f : R N R. We will show next how, in the resent situation, the convergence of the L r -norms with r =, q) of the sums can be deduced from this. The main idea is contained in the next lemma. Lemma Suose ϕ, ϕ 1,... and ϕ are R N -valued random variables such that for all bounded continuous functions f : R N R we have lim Efϕ k) = Efϕ). k
10 38 3 Sums of indeendent random variables Let Φ : R N R be a Borel function such that su k 1 E Φϕ k ) < and E Φϕ) <. If g : R N R is a continuous function satisfying gt) ct) Φt), t R N, where c : R N R is a bounded function satisfying lim t ct) =, then lim Egϕ k) = Egϕ). k Proof. Let g R := g 1 { g <R} + R 1 {g R} R 1 {g R} denote the truncation of g at the levels ±R. By assumtion we have Furthermore, by dominated convergence, lim Eg Rϕ k ) = Eg R ϕ). 3.5) k lim Eg Rϕ) = Egϕ). 3.6) R Fix ε > and choose R > so large that su t >R ct) < ε. Choose R 1 > so large that gt) > R 1 imlies t > R. Then, for all R R 1, su k E gϕ k ) g R ϕ k ) su E1 { g >R} ϕ k ) gϕ k ) ) k su E1 { g >R} ϕ k ) ct) Φϕ k ) ) k ε su E Φϕ k ), k Combined with 3.6) and 3.5), this gives the desired result. 3.7) Now we can finish the roof of Theorem 3.1: Lemma With the notations of Theorem 3.1, for all 1 r < and x 1,...,x N E we have lim E N k ϕ k) n x n r r = γ n x n, where γ 1,...,γ N are indeendent standard Gaussian variables. Proof. Without loss of generality we may assume that max 1 n N x n 1. We fix 1 r < and check the condition of Lemma 3.13 for the functions Φ : R N R and g : R N R defined by N Φt) := ex ) t n x n, gt) := where ϕ k := ϕ k) 1,..., ϕk) N ) and ϕ := γ 1,...,γ N ). r t n x n,
11 If ϕ is a symmetric real-valued random variable, then Eex ϕ ) = Eex 1 {ϕ<} ϕ) + Eex1 {ϕ } ϕ) Hence, since max 1 n N x n 1, EΦϕ k ) N = N N = N O 3.3 Exercises 39 = Eex1 { ϕ<} ϕ) + Eex1 {ϕ } ϕ) Eexϕ). Eex ϕ k) n ) N k j= k N Eexϕ k) n ) Eex r nk+j k ) = N 1 ex 1 k ) + 1 ex 1 k ) ) kn ) kn = N exn/) O1) as k. 3.3 Exercises 1. Let X n ) N be a sequence of indeendent symmetric E-valued random variables, and let r n ) N be a Rademacher sequence which is indeendent of X n ) N. Prove that the sequences X n ) N and r n X n ) N are identically distributed. Hint: As in the roof of Theorem 3. it may be assumed that X n ) N and r n ) N are defined on distinct robability saces. Use Fubini s theorem together with the result of Exercise.. Remark: This technique for introducing Rademacher variables is known as randomisation. It enables one to aly inequalities for Rademacher sums in E to sums of indeendent symmetric random variables in E..!) Let r n ) and r n ) be indeendent Rademacher sequences on robability saces Ω, F, P ) and Ω, F, P ). Prove that on the roduct Ω, F, P) = Ω Ω, F F, P P ), the sequence r m r n ) m, consists of Rademacher variables, but as a doubly indexed) sequence it fails to be a Rademacher sequence that is, the random variables r mr n fail to be indeendent). 3.!) We continue with the notations of the revious exercise. Prove that for 1 < the following version of the contraction rincile holds for double Rademacher sums in the saces L A), where A, A, µ) is a σ-finite measure sace: there exists a constant C such that for all finite sequences f mn ) N m, in L A) and all scalars a mn ) N m, we have m, a mn r m r n f mn C max a mn ) 1 m,n N r m r n f mn m,.
12 4 3 Sums of indeendent random variables Hint: Proceed in three stes: i) the result holds for E = R with exonent ; ii) the result holds for E = R with exonent ; iii) the result holds for E = L A) with exonent. 4. Let 1. A Banach sace E is said to have tye if there exists a constant C such that for all finite sequences x 1,...,x N in E we have )1 r n x n C N x n ) 1. Let q. The sace E is said to have cotye q if there exists a constant C q such that for all finite sequences x 1,..., x N in E we have N x n q) 1 q C q E r n x n ) 1. For q = we make the obvious adjustment in the second definition. Prove the following assertions: a) Every Banach sace has tye 1 and cotye accordingly, a Banach sace is said to have non-trivial tye if it has tye 1, ] and finite cotye if it has cotye q [, )). b) Every Hilbert sace has tye and cotye. c) If a Banach sace has tye for some [1, ], then it has tye for all [1, ]; if a Banach sace has cotye q for some q [, ], then it has cotye q for all q [q, ]. d) Let [1, ]. Prove that if E has tye, then the dual sace E has cotye, = 1. Hint: For each x n E choose x n E of norm one such that x n 1 x n, x n. Then use Hölder s inequality to the effect that for all scalar sequences b n ) N one has N b n ) 1 { N = su a n b n : N a n ) 1 } 1. Remark: The analogous result for saces with cotye fails. Indeed, the reader is invited to check that l 1 has cotye while its dual l fails to have non-trivial tye. 5. Let [1, ]. Prove that a Banach sace E has tye if and only if it has Gaussian tye, that is, if and only if there exists a constant C such that for all finite sequences x 1,..., x N in E we have )1 γ n x n N C x n ) 1.
13 3.3 Exercises 41 Hint: One direction follows from Corollary 3.6. For the other direction use a randomisation argument. Remark: The corresonding assertion for cotye is also true but much harder to rove; see the Notes. Notes. The results of this lecture are classical and can be found in many textbooks. Our resentation borrows from Albiac and Kalton [1] and Diestel, Jarchow, Tonge [35]. Both are excellent starting oints for further reading. The Kahane contraction rincile is due to Kahane [54], who also extended the classical scalar Khintchine inequality to arbitrary Banach saces. It is an oen roblem to determine the best constants K,q in the Kahane- Khintchine inequality; a recent result of Latala and Oleszkiewicz [67] asserts that the constant K,1 = 1 1 is otimal for 1. For a roof of Theorem 3.7 see, e.g., [35]. The roofs of Theorems 3.9 and 3.11 are taken from Albiac and Kalton [1]. The central limit argument in Lemma 3.14 is adated from Tomczak-Jaegermann [1]. The contraction rincile for double Rademacher sums of Exercise 3 has been introduced by Pisier [9]. This roerty, nowadays known under the rather unsuggestive name roerty α) lays an imortant role in many advanced results in Banach sace-valued harmonic analysis. It can be shown that the Rademachers can be relaced by Gaussians without changing the class of saces under consideration. Not every Banach sace has roerty α); a counterexamle is the sace c. The notions of tye and cotye were develoed in the 197s by Maurey and Pisier. As we have seen in Exercise 4, Hilbert saces have tye and cotye. A celebrated theorem of Kwaień [64] asserts that Hilbert saces are the only saces with this roerty: a Banach sace E is isomorhic to a Hilbert sace if and only if E has tye and cotye. Another class of saces of which the tye and cotye can be comuted are the L -saces. For the interested reader we include a roof that the saces L A), with 1 < and A, A, µ) σ-finite, have tye min{, }. A similar argument can be used to rove that they have cotye max{, }. Let f 1,..., f N L A) and ut r := min{, }. Using the Fubini theorem, the scalar Kahane-Khintchine inequality, the tye inequality, Hölder s inequality, and the triangle inequality in L r A), we obtain
14 4 3 Sums of indeendent random variables r n f n L A) ) 1 = A E r n f n ξ) dµξ) A ) 1 K, E r n f n ξ) ) A N = K, f n ξ) ) dµξ) K, A = K, N N K, N f n r 1 r ) 1 dµξ) ) 1 f n ξ) r) ) 1 r dµξ) L r A) f n r L r A) N = K, f n r L A) An alication of the Kahane-Khintchine inequality for L A) to relace the L -moment in the left hand side by the L -moment finishes the roof. It was noted in Exercise 4 that if E has tye, then E has cotye where = 1) and that the analogous duality result for cotye fails. It is a dee result of Pisier [93] that if E has cotye q [, ) and non-trivial tye, then E has tye q, 1 q + 1 q = 1. The fact that a Banach sace has cotye q if and only if it has Gaussian cotye q can be deduced from a dee result of Maurey and Pisier see [1, Chater 11]) which gives a urely geometric characterisation of tye and cotye. For the details we refer to [35]. ) 1 r ) 1 r.
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