On the Equivalence Between Geometric and Arithmetic Means for Log-Concave Measures

Convex Geometric Analysis MSRI Publications Volume 34, 1998

RAFA�L LATA�LA

Abstract. LetX be a random vector with log-concave distribution in some Banach space. We prove that�X� p ≤C p�X�0 for anyp> 0, where p 1/p �X�p = (E�X� ) ,�X� 0 = expE ln�X� andC p are constants depending only onp. We also derive some estimates of log-concave measures of small balls.

Introduction. LetX be a random vector with log-concave distribution (for precise deﬁnitions see below). It is known that for any measurable seminorm and p, q > 0 the inequality

�X�p ≤C p,q�X�q holds with constantsC p,q depending only onp andq (see [4], Appendix III). In this paper we show that the above constants can be made independent ofq, which is equivalent to the inequality

�X�p ≤C p�X�0, (1) where�X� 0 is the geometric mean of�X�. In the particular case in whichX is uniformly distributed on some convex compact set inR n and the seminorm is given by some functional, inequality (1) was established by V. D. Milman and A. Pajor [3]. As a consequence of (1) we prove the result of Ullrich [6] concerning the equivalence of means for sums of independent Steinhaus random variables with vector coeﬃcients, even though these random-variables are not log-concave (Corollary 2). To prove (1) we derive some estimates of log-concave measures of small balls (Corollary 1), which are of independent interest. In the case of Gaussian random variables they were formulated and established in a weaker version in [5] and completelely proved in [2].

123 124 RAFA�L LATA�LA

Deﬁnitions and Notation. LetE be a complete, separable, metric vector space endowed with its Borelσ−algebraB E.Byµ we denote a log-concave probability measure on (E,B E) (for some characterizations, properties and examples, see [1]) i.e. a probability measure with the property that for any Borel subsets A, B and all 0<λ< 1 we have − µ(λA + (1−λ)B)≥µ(A) λµ(B)1 λ.

We say that a random vectorX with values inE is log-concave if the distribution ofX is log-concave.For a random vectorX and a measurable seminorm�.� on E (i.e. Borel measurable, nonnegative, subadditive and positively homogeneous function onE) we deﬁne

p 1/p �X�p = (E�X� ) forp>0 and

�X�0 = lim �X�p = exp(E ln�X�). p→0+ Let us begin with the following Lemma from [1]. Lemma 1. For any convex, symmetric Borel setB andk≥1 we have

(k+1)/2 1−µ(B) µ((kB)c)≤µ(B) . µ(B) � � Proof. The statement follows immediately from the log-concavity ofµ and the inclusion k−1 2 B+ (kB)c ⊂B c. k+1 k+1 � Lemma 2. IfB is a convex, symmetric Borel set, withµ(KB)≥ (1 +δ)µ(B) for someK>1 andδ>0 then

µ(tB)≤ Ctµ(B) for anyt∈ (0, 1),

whereC=C(K/δ) is a constant depending only onK/δ. Proof. Obviously it’s enough to prove the result fort=1/2n,n=1,2,... . So let usﬁxn and deﬁne, foru≥ 0,

Pu ={x:�x� B ∈(u−1/2n, u+1/2n)}, where

�x�B = inf{t >0:x∈ tB}. −1 By simple calculationλP u + (1−λ)(2n) B⊂P λu, so

λ −1 1−λ µ(Pλu)≥µ(P u) µ((2n) B) forλ∈ (0, 1). (2)

From the assumptions it easily follows that there existsu≥ 1 such thatµ(P u)≥ δµ(B)/Kn. Letµ((2n) −1B)=κµ(B)/n. Ifκ≤2δ/K we are done, so we will GEOMETRIC AND ARITHMETIC MEANS FOR LOG-CONCAVE MEASURES 125

assume thatκ≥2δ/K. Then by (2) it follows thatµ(P 1)≥δµ(B)/Kn. The −1 setsP (n−1)/n,P(n−2)/n,...,P1/n, (2n) B are disjoint subsets ofB, and hence −1 µ(B)≥µ(P (n−1)/n) +···+µ(P 1/n) +µ((2n) B).

−1 Using our estimations ofµ(P 1) andµ((2n) B) we obtain by (2) − − − µ(B)≥n 1µ(B)((δ/K) (n 1)/nκ1/n +···+(δ/K) 1/nκ(n 1)/n +κ) = κ 1−δ/Kκ κ 1 = µ(B) ≥ µ(B) . n 1−(δ/Kκ) 1/n 2n 1−(δ/Kκ) 1/n Therefore κ≤2n(1−(δ/Kκ) 1/n)≤ 2 lnKκ/δ, so thatκ≤C(K/δ) and the lemma follows. �

Corollary 1. For eachb<1 there exists a constantC b such that for every log-concave probability measureµ and every measurable convex, symmetric set B withµ(B)≤b we have

µ(tB)≤C btµ(B) fort∈ [0, 1]. Proof. Ifµ(B)=2/3 then by Lemma 1µ(3B)≥5/6 = (1 + 1/4)µ(B), so by Lemma 2 for some constant C˜1,µ(tB)≤ C˜1tµ(B). Ifµ(B)∈ [1/3,2/3] then obviouslyµ(tB)≤2 C˜1tµ(B). Ifµ(B)<1/3, let K be such thatµ(KB)=2/3. By the above caseµ(B)≤ −1 C˜1K µ(KB), and hence µ(KB) K≤2 C˜ −1 . 1 µ(B) � � So Lemma 2 gives in this case thatµ(tB)≤ C˜2tµ(B) for some constant C˜2. Finally ifµ(B)>2/3, butµ(B)≤b< 1 then by Lemma 1 for someK b <∞, −1 µ(Kb B)≤2/3 and we can use the previous calculations.�

Theorem 1. For anyp>0 there exists a universal constantC p, depending only onp such that for any sequenceX 1,...,Xn of independent log-concave random vectors and any measurable seminorm�.� onE we have n n Xi ≤C p Xi . � i=1 �p � i=1 �0 � � � � � � � � � � Proof. Since a convolution� of log-concave� � measures� is also log-concave (see [1]) we may and do assume thatn = 1. Let

M = inf{t:P(�X 1� ≥t)≤2/3}. Then by Lemma 1 (used forB={x∈E:�x� ≤M}) it follows easily that �X1�p ≤a pM forp> 0 and some constantsa p depending only onp. By similar reasoning Corollary 1 yields�X 1�0 ≥a 0M. � 126 RAFA�L LATA�LA

Corollary 2. LetE be a complex Banach space andX 1,...,Xn be a sequence of independent random variables uniformly distributed on the unit circle{z∈C: |z|=1}. Then for any sequence of vectorsv 1, . . . , vn ∈E and anyp>0 the following inequality holds:

vkXk ≤K p vkXk , p 0 �� whereK is a constant� depending� only onp�. � p � � � � Proof. It is enough to prove Corollary forp≥ 1. LetY 1,...,Yn be a sequence of independent random variables uniformly distributed on the unit disc{z: |z|≤1}. By Theorem 1 we have

vkYk ≤C p vkYk . (3) p 0 �� But we may representY � in the form� Y =�R X ,� whereR are independent, �k � k � k k � k identically distributed random variables on [0, 1] (with an appropriate distribution), which are independent ofX k. Hence, by taking conditional expectation we obtain

vkYk ≥(ER 1) vkXk . (4) p p �� Finally let us observe� that for any� u, v∈E the� function�f(z) = ln�u+zv� is � � � � subharmonic onC, sog(r) =E ln�u+ rvX 1� is nondecreasing on [0,∞) and therefore

vkXk ≥ vkYk . (5) 0 0 �� The corollary follows from� (3), (4) and� (5).� � � � � � � Acknowledgements

This paper was prepared during the author’s stay at MSRI during the spring of 1996. The idea of the proof of Corollary 2 was suggested by Prof. S. Kwapie´n.

References

[1] C. Borell, “Convex measures on locally convex spaces”, Ark. Math. 12 (1974), 239–252. [2] P. Hitczenko, S. Kwapie´n,W. V. Li, G. Schechtman, T. Schlumprecht, and J. Zinn, “Hypercontractivity and comparison of moments of iterated maxima and minima of independent random variables”, Electron. J. Probab.3 (1998), 26pp. (electronic).’ [3] V. D. Milman and A. Pajor, “Isotropic positions and inertia ellipsoids and zonoids of the unit balls of a normedn-dimensional space”, pp. 64–104 in Geometric aspects of functional analysis: Israel Seminar (GAFA), 1987-88, edited by J. Lindenstrauss and V. D. Milman, Lecture Notes in Math. 1376, Springer, Berlin 1989. [4] V. D. Milman and G. Schechtman, Asymptotic theory ofﬁnite dimensional normed spaces, Lecture Notes in Math. 1200, Springer, Berlin, 1986. GEOMETRIC AND ARITHMETIC MEANS FOR LOG-CONCAVE MEASURES 127

[5] S. Szarek, “Conditional numbers of random matrices”, J. Complexity7 (1991), 131–149. [6] D. Ullrich, “An extension of Kahane–Khinchine inequality in a Banach space”, Israel J. Math. 62 (1988), 56–62.

Rafa�lLata�la Institute of Mathematics Warsaw University Banacha 2 02-097 Warsaw Poland [email protected]