Two elementary probabilistic proofs of the Kantorovich inequality are given and various generalizations and inequalities are discussed

Bühler, Wolfgang J.

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Rev..0~.ta Co.f.omb..(.a.n.a. de Ma.tema..uc.a.6Vol. XXI (1987), ptfg-6. 147-154TWO PROOFS OF THE KANTOROVICH INEQUALITYAND SOME GENERALIZATIONSbyWol fgang J. BUHLERAbstract. Two elementary probabilistic proofs of the Kan-torovich inequality are given and various generalizations andinequalities are discussed.§l. Introduction. The Kantorovich inequality in its simplestform seems to have been found originally as early as 1914 bySchweitzer [8]. In the context of statistical applicationsit is usually stated in the form1 ~ -1 Z(x 'Ax) •(x'A x) ~ (a+ b) /4ab (1 )where A is a positive definite nxn matrix with eigenvalueso < a A1 ~ AZ ~ ...~An = b and x is a unit vector.For 'given A there are unit vectors xl' Xz for which theleft or right inequality respectively is an equality. Therehave been various generalizations and modifications, see e.g. [6] and t he references given there. My own interest inthe matter comes from reading the proof in [7]. This proof- 1 .starts by transforming the term F = (x,Ax) (x'A x) by dlago-147nalizing A inton 2 n 2 -1 . n 2F = (L YiAi)( L y.A. ),Wlth L y. 1.i=l i=l 1 1 i=l 1Letting p. = y? this has the following probabilistic inter-1 1pretation not used in [7]. F is the product of the expectedvalue of a random variable Z taking values A. with probabil-1ities Pi and of the expected value of liZ. Both proofs giv-en in the present paper use the representation of F asEZE(l/Z).§2. First proof and generalization. Let Z be a random vari-able, not necessarily discrete, taking values in [a,b], 0 <a < b < 00THEOREM 1. 1 ~ E Z • E ( 1 I Z) -s ( a + b) 2 14 a b . ( 2)The ~igh~ inequali~y i~ an equali~y i6 and only 16 Z take6eaQh 06 ~he value6 a and b wi~h p~obabili~y 1/2, the le6tinequal~~y i6 and only i6 Z i6 degene~a~e.REMARK. The special case whereis defined as an integrable functioncontained in Schweitzer [8J.the random variable Zon [O,lJ is already\Proof of Theorem 1. We start from the well known boundfor the covarianceEZ·E(l/Z) ~ E[Z(l/Z)] + [V(Z)V(l/Z)]~the symbol V denoting the variance. Combining this with thefact that the variance of the bounded variable Z is at most(a+b)2/4 (and similarly for V(l/Z)) we obatin the upper bound1+{(a+b)2/4'(1/a+l/b)2/4}~ = (a+b)2/4ab. The first bound isattained if liZ is a.s. a decreasing linear function of Zwhich allows at most two values, the second if the two val-148ues a and b have probability 1/2 each. The left inequalityin (2) could be proved via essentially the same path. It ishowever also a simple application of Jensen's inequality.COROLLARY. LeA: (II,F,ll) be a mea.6ulte .6pac.e with ll(rl)< 1and let f be a mea.6ultable Iteal valued 6unc..:tion on II wi.:tha < few) < b almo.6t eveltywhelte Illi. ThenJfdll f(l/f)dll < (a+b)2/4ab. (3)Proof. Augment II by an additional point 0 and II by apoint mass 1-11(1I) at 0 to make II into a probability space.Define the random variable Z as f on II and Z(o) = a, say.Then apply Theorem 1.Let x, y be two unit n-vectors, A a matrix as above(without loss of generality assumed to be diagonal). ThenllUi}) = IXiYil defines a measure on {1,2, ...n} to which wecan apply the Corollary with f(i) = \-. The resulting ine-1quali ty(x'Ay).(y'A-1x) ~ '\.lx.Y.Il:(l/\.)\x.y.1 ~ (a+b)2/4abL 1 11 1 11is Strang's [9J generalization of Kantorovich's inequality.§3. Second proof and further generalization. The second proofmakes use of the convexity of f(x) = l/x and can in fact beapplied for any convex f. We are thus led toTHEOREM 2. Le.:to < a < b < 00; le.:t fZ be a Itan.dombe de6ined onThen we havevaltiabte with P (a~ Z~ b) = 1,[a,b] and c.onvex wi.:thf(a) = A > B feb) .EZ-Ef(Z)~ max{(Ab-Ba)2/[4(A-BHb-a)), aA, bBl. (4)Equality i.6 Iteac.hed 60lt .60me Z taQing at mO.6t .:the .:two vatu~a ilnd b. 16 f if.> c.one-ave .:then .:the inequality if.> Iteveltf.> ed, i.e.149(4) yield4 a lowe~ bound.Proof. The two dimensional random variable (Z,f(Z))takes its values on the graph of f. E being a convex opera-tion the point (EZ,Ef(Z)) must lie in the (closed) shadedregion R bounded by the graph of f and by the straight lineg(z) = A+(B-A)(z-a)/(b-a).yl---'---~B _.- --- -I- -- - - --a z bThe aim is now to maximize the area zy of a rectangle with(z,y) , R. Obviously zy' > zysuch that we can limit oursearch to points (z,g(z)). The area h(z) = z e g Iz) is maxi-mized when-0 = h'(z) = g(z) + zg'(z) = {A(b-a)+ (B-A)(z-a)+ z(B-A)}/(b-a) ,i.e. when z = (Ab-Ba)/2(A-B)(Ab-Ba)2/[4(A-B)(b-a)]. Here Zoa = (Ab+Ba-2Bb)/[2(b-a) (A-B)].Zo with h(zo) == aa + (l-a)b withThe local maximum of h at z being the only extremum it isoobviously a global maximum over the whole line g. Howevernot always is a ~ z ~ b. If it is, i.e. if 0 ~ a. ~. 1 thenothe variable Z with P(Z=a) = a = l-P(Z=b) will have EZEf(Z)= h(z ); if a. < 0 the Z degenerate at a will bring EZ-Ef(Z)o= aA; if a > 1 then Z degenerate at b will attain the upperbound in the form bB.150REMARK. If in the figure we let z = EX = Sb+(l-S)aand y = Ef(X) for some random variable X wi th P(a.::;X.::;b) = 1then with y' = BB + (l-S)A we find the estimatef(EX) ~ Ef(X) ~ [(Ab-aB) - (A-B)EX] /(b-a).The left inequality is of course Jensen's, the right ine-quality holds without the restrictions that 0 < a and A > B.In the present context, if EX is known, it is sharper than(4) .COROLLARY. Let Z be a~ in the Theo~em. Let f,g both beeonvex, o~e 06 them ine~ea~ing the othe~ dee~ea~ing, f(a)A,f(b) = B, g(a) = C,g(b) = D. ThenEg(Z) Ef(Z) ~ max{(AD-BC)2/[4(C-D)(B-A)],AC,BD} (5)Ppoof. Under the conditions stated fog-1 is convex anddecreasing. Thus the theorem can be applied with Y = g-l(Z)in lieu of Z. (4) then takes the form (5).§4. The inequality of Greub and Rheinboldt. The followinginequality contains that given by Greub and Rheinboldt [3]as a special case.THEOREM 3. Let X,Y be ~andom va~iabte~ de6ined on ap~obabitity ~paee (~,F,P) with a ~ X .::;b and A .::;Y ~ B a.s.Then(6)Proof: We first remark that both sides of (6) are in-variant to scale. We thus rescale X and Y to achieve EXY = 1wi thout changing the ratio X/Y. The equation !XYdP = EXY = 1displays XY as the density of a probability measure Q. De-noting by EQZ = jZdQ = jZXYdP = EZXY the corresponding ex-pectation we obtain2 2 (a/Btb/A~2 _ (aA+bB)2EX ·EY = EQ(X/Y)'EQ(Y/X) ~ 4(a/B)(b N- 4aAbB151where we have applied (2) to Z(n,F,Q).X/Y as a random variable onCOROLLARY. I6 G ~nd H ~4e eommu~ing m~~4iee~ wi~h ei-genv~tuec 0 = a = A1 ~ A2~' ..~An =b ~nd 0 < A ,< ~1 ~ ~2 ~... ~ ~n = B, ~hen(7)Proof. G and H can simultaneously be diagonalized.The Corollary then, in an analogous way as (1), followsfrom Theorem 1.§5. Examples. Let us go back to the original situation of amatrix A with eigenvalues 0 < a = A1~ ...~An = b. Then form,k > lone hasfor all x of unit length. This is just an applicatioR of thecorollary of Section 3 with fez) = zm,g(z) = z-k Except inh k th 't t' 'Whl'ch am-k or bm-kt e case m = ere are Slua Ions Inis the appropriate upper bound. When m = 1 or k = 1 it maybe worthwhile to cancel two factors b-a to get the bound e.g. 'in the form2 -1 2 2 2(x'A x)(x'A x) ~ max{(a,b,(b +ab+a ) 1[4ab(b+a)]} u (8)As a further illustration we consider an applicationof Theorem 2. Let 0 < s < r and let X be a random variablewith Pea ~ X ~ b) 1. Put Z = XS and fez) = z-rls. Thenaccording to Theorem 2 we havev. (9)Considering the special situation in which EXs = 1 thisleads to (EXs) l/s(EX-r) 1/r = (EXs EX-r)2/r ~ U where152For the general situation EXs =: CS the upper bound for(EXs) l/s(EX-r) l/r will be Uc(r-s)/r which is at mostUb(r-s)/r.Our approach also covers a more general situation. LetH be a Hilbert space of real valued square integrable func-tions with scalar product <x,Y> = !x(t)y(t)dt. Let the oper-ator A be given by (Ax) (t) = h(t)x(t) where 0 < a ~ h(t) ~ bfor all t. Then (A-lx)(t) = x(t)/h(t) and the value U givenin (8) will be an upper bound for the product <A2x,xXA-1x,x>,a result which obviously extends to the class of self ad-joint integral operators that can be transformed into the"diagonal" form considered. Similarly the inequalities ofStrang and of Greub and Rheinboldt apply in this context.REFERENCES[ 1 ][21[ 3]Beckenbach, E.F., O~ the ~~equai~ty 06 Ka~to~ov~Qh. Mer.Math. Monthly, 71, (1964) 606-619.Clausing, A., Ka~to~ov~Qh-type ~~equai~t~e~. Amer. Math.Monthly, 89 (1982) 314 and 327-330.Greub, W. and Rheinboldt, W., O~ a ge~e~ai~zat~o~ 06 a~~~equai~ty 06 L.V. Ka~to~ov~Qh. Proc. Amer. Math.Soc., 10 (1959) 407 - 4 15 .Kantorovich, L.V., FU~Qt~onai anaiy~~~ and appi~ed mathe-mat~Q~. Uspehi Mat. Nauk., 3 (1948) 89-135 (alsotranslated from the Russian by C.D. Benster, Nat.Bur. Standards Report N~ 1509, March 7, 1952).(l948) .Khatri, C.G. and Rao, C.R., Some exten~~on~ 06 Kanto~o-V~Qh ~nequai~ty and ~tat~~t~Qai appi~Qat~o~~. J.Mul t i,va r , Anal., 11, (1981) 498 - 505.Rao, C.R. and Khatri, C.G., Some qe~e~ai~zat~on~ 06 Ka~-tO~OV~Qh ~nequai~ty. Sankhyl, Ser. A, 44 (1982) 91-102.Raghavachari, M., A i~nea~ p~og~a~m~~g p~o?6 ?6.Kanto~0-v~Qh I ~ .lnequai~ty. The Ame ri can StatIstIcIan, 40(1986) 136-137.[4][51[6][ 7]153[8 J Schweitzer, P., An inequali~y eonee~ning ~he a~i~hme~iemean (in Hungarian). Math. Phys. Lapok, 23 (1914)257-261.Strang, W.G., On ~he Kan~o~ovieh inequali~y. Proc. Amer.Math. Soc,, 11 (1960) 468.[9J*FaehbVLueh M~e.rm;tik.JohanneA GutenbVl.g-Univ~itat0-6500 MainzRepubliea FedVLal de Alemania.(Recibido en octubre de 1986).,154

Two proofs of the kantorovich inequality and some generalizations

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Two proofs of the kantorovich inequality and some generalizations

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