A converse Gaussian Poincare-type inequality for convex functions

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1 Statistics & Proaility Letters A converse Gaussian Poincare-type inequality for convex functions S.G. Bokov a;, C. Houdre ; ;2 a Department of Mathematics, Syktyvkar University, 6700 Syktyvkar, Russia Southeast Applied Analysis Center, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30322, USA Received August 998; received in revised form Novemer 998 Astract A converse Poincare-type inequality is otained within the class of smooth convex functions for the Gaussian distriution. c 999 Elsevier Science B.V. All rights reserved MSC primary 60E5; 60E99; secondary 26D0 Keywords Poincare-type inequalities; log-soolev inequalities; Gaussian distriution These notes form a companion to a previous paper of the authors where the question of otaining converse Poincare-type inequalities was investigated. By a Poincare-type inequality for a given proaility measure on R, one usually means an analytic inequality of the form Var f6k f 2 2; which relates the -variance of an aritrary smooth function f on R to the L 2 -norm of its derivative f, and where the constant K does not depend on f. For example, as rst noted in Klaassen 985, the aove inequality is satised y the doule exponential distriution with the optimal constant K = 4. In fact, as already mentioned in Borovkov and Utev 983, if is a random variale with d.f. F and density p such that, for some x 0 ; Fx6cpx, if x x 0, and Fx6cpx, if x6x 0, then Eq. holds with K =4c 2. For the doule exponential measure, Eq. can e inverted in the class of all convex f, in the sense that for some K 0, Var f K f 2 2; Corresponding author. Tel ; fax address houdre@math.gatech.edu C. Houdre Research supported in part y the Russian Foundation for Fundamental Research. This author greatly enjoyed the hospitality of the School of Mathematics, Georgia Institute of Technology, Atlanta, where part of this work was carried out. 2 Research supported in part y the NSF Grant No. DMS This author greatly enjoyed the hospitality of CIMAT, Gto, Mexico where part of this work was carried out /99/$ - see front matter c 999 Elsevier Science B.V. All rights reserved PII S X

2 282 S.G. Bokov, C. Houdre / Statistics & Proaility Letters see Bokov and Houdre, 997a. Actually, such a converse convex Poincare-type inequality holds for many other symmetric proaility measures with suciently heavy tails. It is however not satised y the canonical Gaussian measure =, and at the same time, the inequality for = holds true with K =. Thus, the classical Gaussian Poincare-type inequality is relatively rough, and for a suitale invertile estimate of Var f, we have to look for a quantity somewhat dierent from f 2 2. As it turns out, the L 2 -norm of the derivative can e replaced y a suitale Orlicz norm Theorem. There exist positive numerical constants K 0 and K such that; for every convex function f R R; K 0 f 2 6Var f6k f 2 ; 2 where is the Orlicz norm in L with x=x 2 =[log + x]; x 0. Recall that the Orlicz norm of a measurale function u in L is dened y { } u u = inf 0 E 6 ; where the aove expectation is with respect to. One easily veries that aove is a Young function, i.e., it is positive, convex, increasing in x 0, with 0 = 0, so that is indeed a norm. The inequality on the right in 2 is not really new. It holds true for all asolutely continuous f, without the convexity assumption, and is a consequence of an inequality of Talagrand 994, Theorem 6 onthe discrete cue for every function f {0; } n R, n Var n p f6kp i f 2 3 i= Here, the variance and the Orlicz norm are understood with respect to the n-fold tensor product p n of the Bernoulli measure on {0; } with parameter p 0;, and i f denotes the increment of f along the ith coordinate. Applying the aove discrete inequality with p = =2, for deniteness to the functions f n x=f2x + + x n n= n; x {0; } n, with f smooth enough, and letting n, we otain y the central limit theorem the right inequality in 2 with K =4K=2. In addition to the proof of Eq. 3 given in Talagrand 994, it is also noted there that this inequality could also e otained, y duality, from Gross 976 logarithmic Soolev inequality on the discrete cue in the case p ==2. For an aritrary proaility metric space it is shown, in the appendix at the end of this paper, that the right inequality in 2 implies a log-soolev inequality. However, we do not know if the converse statement is true in general, i.e., we do not know if a log-soolev inequality implies the right inequality in 2. In this connection, it might e worthwhile to note that, in contrast to 2, the Gaussian log-soolev inequality cannot e inverted in the class of all convex functions. Indeed, if we apply E f 2 logf 2 E f 2 loge f 2 KE f 2 ; to the functions e f ; 0, with f convex and then let 0, we arrive at 2Var f KE f 2, which is clearly false. We can now pass to the proof of the left inequality in 2. To denote the variance, we also write Varf; instead of Var f, and Var instead of Varf; when the function is identity fx=x. We also write E f to denote the expectation of f with respect to. As usual, x=;x] = x e t2 =2 dt; x R; 2 denotes the distriution function of, and == 2e x2 =2 denotes its density.

3 S.G. Bokov, C. Houdre / Statistics & Proaility Letters It is much easier to prove the left inequality in 2 assuming additionally that f is monotone. To treat the general case, we will have to consider an inequality such as Varf; f 2 dq 4 R in the class of all convex functions f on R for a suitale nite positive symmetric measure Q on R, asolutely continuous with respect to. Before starting, let us note one sucient condition for Eq. 4. First, we can assume that for some a R; fis non-increasing on ;a] and is non-decreasing on [a; +. Then, writing = a a + a + a ; where a and + a are respectively the left and the right conditional restriction of to the half-lines ;a] and [a; +, i.e., we have a A= A ;a] ; + a A= a A [a; + ; A R; Borel; a Varf; =avarf; a + avarf; + a +a ae + a f E a f 2 avarf; a + avarf; + a Therefore, Eq. 4 will follow from Varf; a a f 2 dq; a 5 Varf; + a f 2 dq 6 a a Since Eqs. 5 and 6 are equivalent y the symmetry of and of Q, and since f is non-decreasing on [a; +, in order to get Eq. 4, it will e sucient to estalish Eq. 6 in the class F + of non-decreasing convex functions on R. To do so, we use Lemma. Let e a random variale with nite second moment; and continuous distriution function; and let Q e a nite positive measure on R. The following are equivalent a Covf;g R f g dq; for all f; g F + ; such that Ef 2 and Eg 2 are nite. Varf R f 2 dq; for all f F +. c Var a + Q[a; + ; for all a R. When Q is the law of, the aove statement is Lemma in Bokov and Houdre 997a, ut the proof there extends to aritrary Q. Just note that the functionals f; g Covf;g; f; g f g ; are ilinear, and that the function a Cov a + ; + is non-decreasing in a ;] note also the fact that any f F + is a mixture of functions f a x=x a +. Now, y Lemma, the inequality 6 for the class F + is equivalent to the same inequality with fx= x + ; a, that is, to Varx + ; + a a dq = Q[; + a ;

4 284 S.G. Bokov, C. Houdre / Statistics & Proaility Letters that is, to Varx + ; + a Q[; + + a [; + 7 Let us recall another elementary statement which was oserved in the proof of Theorem in Bokov and Houdre 997a and which is needed here only for =. Lemma 2. Let e a proaility measure on R such that [; + 0; for all R. Then; for all a6; Varx + ; a + a + Varx + ; + [; + = Var+ In order words, the left-hand side of the aove inequality is minimized for a =. Using Lemma 2, the left-hand side of 7 can e estimated from elow y Var +, so that Eq. 7 and therefore Eq. 4 will follow from Var + + Q[; Thus, the measure Q in Eq. 4 can e chosen via the identity Q[; + = Var + 8 Since Var + = Varx + ; + 2 = E x +2 E x + ; the equality 8 is just Q[; + = E x +2 This leads us to Lemma 3. For all convex functions f on R; Var f K f x 2 +x 2 dx E x Proof. As it follows from the aove discussion, it only remains to show that, for all R, Q[; + K +x 2 dx; where K is a universal constant and where the function Q is dened via Eq. 8. For =, the aove inequality is trivial since, y Eq. 8, Q; + =Var=; so only the ehavior, as +, needs to e considered on oth sides of the aove inequality. That is we just need to compute the correct asymptotics the right-hand side of Eq. 9, as +. We start with the second term there. Integrating y parts, we have E x + = x dx =

5 Now, Hence, = E x and thus But, since E x it follows that S.G. Bokov, C. Houdre / Statistics & Proaility Letters x 4 3 +O +O +O 2 ; 5 ; E x O Let us estimate the rst term on the right-hand side of Eq. 9 E x +2 = =2 2 =2 x 2 dx dx x x dx x x x 3 +O x 5 dx [ x 2 x + x ] x 3 +O 4 dx Denote y A this last integral and proceed to estimate every single one of its components. [ I dx = = ] 3 +O 5 II x 2 dx = d = x3 3 +O 5 III x dx = x 2 d = 2 2 x 3 dx

6 286 S.G. Bokov, C. Houdre / Statistics & Proaility Letters Hence, IV thus, Finally, V x = 2 +2 x 4 d = 2 dx = 2 4 +O O 5 + x 3 dx = d = x4 4 +O x 3 dx = 3 +O x 4 dx =O 5 5 Comining these ve estimates, we have [ A = ] 3 +O 5 = 3 +O 2 Hence, E x Q[; K 2 +O 2 ; 3 +O 4 +O 6 or K x 2 dx ; as +. This nishes the proof of Lemma 3. ; To continue, recall that given a Young function N, the corresponding Orlicz norm is dened y { } u u N = inf 0 EN 6 0 Lemma 4. For all measurale functions u; v on some proaility space; Euv 63 u x log +x v e x

7 S.G. Bokov, C. Houdre / Statistics & Proaility Letters Proof. One may assume u; v 0. By homogeneity, let v ex = so that y Eq. 0 Ee v = 2. But 2u sup Euv = Eu log ; Ee v =2 Eu which is just a well-known functional representation of the entropy. Thus to prove the result, it is enough to show that for all u 0 such that Eu 0, 2u Eu log 63 u x log +x Eu Again, y homogeneity, let u x log +x =, that is let Eu log + u= By Jensen s inequality, Eu log + u Eu log + Eu, so Eu log + Eu6 Hence Eu62, since 2 log 2 and since the function x log + x is increasing. Therefore, using also that fact that x log x =e for all x 0, we have 2u Eu log = Eu log 2 + Eu log u Eu log Eu Eu 6 2 log 2 + Eu log + u+e and Lemma 4 is proved. =2log2++e 63; Lemma 5. For every measurale function g R R, g 2 6K gx 2 +x 2 dx; where K is universal constant; and is the Orlicz norm in the space L with respect to the Young function x=x 2 =[log + x]; x 0. Proof. By homogeneity, let g =; also we can assume that g 0. Applying Lemma 4 to u= gx=[+x 2 ] and vx=+x 2, we have = gd = 6 3 = C g v v d g v v ex x log+x g v ; x log+x where C =3 v ex + is universal. Thus, g v x log x C = c

8 288 S.G. Bokov, C. Houdre / Statistics & Proaility Letters which is equivalent to saying that g E cv log + g cv Since v, it follows that g E v log + g c c From the very denition of, we thus have g 2 E v log + g log g 2 + c clog + g Now, note for some = c 0, x 2 log + 6 log + x; for all x 0 2 c log + x Indeed, this inequality holds true near the points x = 0 and x =+. Finally, using Eq. 2 in Eq., we get E g 2 =v c=, that is, gx 2 +x 2 dx c Proof of Theorem the left inequality 2. Comine Lemmas 3 and 5 with g = f. Appendix Let M; e a metric space equipped with a Borel proaility measure. For every function f on M, one denes its modulus of gradient fx fy fx = lim sup ; x M; x;y 0 + x; y with the convention that fx = 0, when x is an isolated point in M. We apply this denition to the class A of all functions f on M which have a nite Lipschitz constant on every all in M and in this case the function f is nite and Borel measurale cf. Bokov and Houdre 997 for details. Now for as in Theorem, consider inequalities of the following two types Varf6K f 2 ; 3 Ef 2 log f 2 Ef 2 log Ef 2 6K E f 2 ; 4 where the expectations, the variance and the Orlicz norm are with respect to, where f is an aritrary function in A, and where the constants K and K do not depend on f. Here we prove that Proposition. The inequality 3 implies the inequality 4 with K 6cK; where c is a universal constant. It is proale that the converse statement in the aove proposition is not true. The L -norm is weaker than the L 2 -norm. So, comparing 3 with 4 is very similar to the analogous prolem of comparing the inequality a E f Ef 6KE f, involving the L -norm of the modulus of the gradient, with a Poincare-type inequality E f Ef 2 6K E f 2, involving the L 2 -norm of the modulus of gradient. It is well known that a implies which is a version of a Cheeger s inequality. However, the converse is not true. Already on the real line M =R there exist proaility measures which satisfy ut do not satisfy a with a nite K.

9 S.G. Bokov, C. Houdre / Statistics & Proaility Letters Lemma 6. u 62 u 2 ; for all u L. Proof. It is easy to verify that y6 y +y 2, for all y R. Thus, if u 2 2 Eu2 =, then E u62. Hence, E 2 u6 2E u6 since is convex and 0 = 0. Lemma 7. For all measurale functions u and v; uv 624 u e x 2 v 2. Proof. We use Young s inequality xy6 x 0 tdt + y 0 sds Ax+By; x;y 0; where is an aritrary continuous increasing function on [0; + with 0 = 0; + = +, and where is the inverse of. Set s= 2 log + s so that By6y 2 log + y. Since t=e t2 =2, we also have Ax6xe x2 =2. Thus, xy6xe x2 =2 + y 2 log + y; x;y 0 5 Now, y the very denition of ; a6 2 a, whenever. Hence, using also the convexity of,we have a + 62 a+2, for all a; 0. Applying this inequality to a = xe x2 =2 ;= y 2 log + y, we get from Eq. 5 xy62 xe x2 =2 +2 y 2 log + y; x;y 0 6 Since + xe x2 =2 e x2 =2, we have xe x2 =2 =x 2 e x2 =[log + xe x2 =2 ]62e x2. Setting y =e x2 =2, we also have y 2 log + y = x e x2 =2 6 x e x2 =2 62e x2 = 2 + y 2 These two estimates together with 6 give xy64e x2 = y 2. Therefore, if u; v 0 this can e assumed and u e x 2 = v 2 =, that is, if Ee u2 = 2 and Ev 2 =, then E uv64ee u2 +4E + v This implies the inequality of the lemma. Proof of Proposition. Introduce the functional Lf = sup [Ef + c 2 log f + c 2 Ef + c 2 log Ef + c 2 ]; c R which is dened for all measurale f, is non-negative, and is nite if and only if Ef 2 logf 2 +. Since the modulus of the gradient f is invariant under translations f f + c, the inequality 4 can formally e strengthened and written as Lf6K E f 2. On the other hand, as shown in Bokov and Gotze 999, Proposition 4, 2 3 f Ef 2 N 6Lf6 5 2 f Ef 2 N ; 7 where the Orlicz norm is generated y the Young function Nx =x 2 log + x 2. Thus, up to a constant, Eq. 4 is equivalent to the inequality f Ef 2 N 6K E f 2 ; 8 where N is as aove. To deduce Eq. 8 from Eq. 3, we may assume that f is ounded, Ef = 0 and E f 2 ==K and that K 0. Apply Eq. 3 to the function g=f log + f 2. Since x log + x 2 62 log + x 2 and thus g 62 log + f 2 f, we get ENf =Ef log + f Ef log + f K 2 log + f 2 f 2 9

10 290 S.G. Bokov, C. Houdre / Statistics & Proaility Letters Once more y Eq. 3, applied to f, and y Lemma 6, Ef 2 6K f 2 64K f 2 2 =4 Hence, y Cauchy Schwarz and Jensen s inequalities, Ef log + f 2 2 6Ef 2 E log + f 2 6Ef 2 log + Ef 2 64 log 5 This gives a ound for the rst term in Eq. 9. In order to estimate the second term, note that, for the Young function x =e x2, we have according to Eq. 20 that E log + f 2 = Ef 2 64, so that log + f Hence, y Lemma 7, log + f 2 f 624 log + f 2 f 2 696= K Comining these ounds, we otain from Eq. 9 that ENf64 log Hence, f N 6200, since N tx6t 2 Nx for t 6. This proves Eq. 8 with K = K. By the second inequality in 7, we nally get Lf K. Thus, Proposition is proved with c = References Bokov, S.G., Gotze, F., 999. Exponential integraility and transportation cost related to logarithmic Soolev inequalities. Bielefeld University, Preprint SFB ; J. Func. Anal., in press. Bokov, S.G., Houdre, C., 997a. Converse Poincare-type inequalities for convex functions. Statist. Proa. Lett. 34, Bokov, S.G., Houdre, C., 997. Some connections etween isoperimetric and Soolev-type inequalities. Mem. Amer. Math. Soc Borovkov, A.A., Utev, S.A., 983. On an inequality and a related characterization of the normal distriution. Theory Proa. Appl. 28, Gross, L., 976. Logarithmic Soolev inequalities. Amer. J. Math. 97, Klaassen, C.A.J., 985. On an inequality of Cherno. Ann. Proa. 3, Talagrand, M., 994. On Russo s approximate zero-one law. Ann. Proa. 22 3,

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