MULTINOMIAL PROBABILITIES, PERMANENTS AND A CONJECTURE OF KARLIN AND RINOTT R. B. BAPAT

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 102, Numbe 3, Mach 1988 MULTINOMIAL PROBABILITIES, PERMANENTS AND A CONJECTURE OF KARLIN AND RINOTT (Communicated R. B. BAPAT by Bet E. Fistedt) ABSTRACT. The pobability density function of a multipaamete multinomial distibution can be expessed in tems of the pemanent of a suitable matix. This fact and cetain esults on conditionally negative definite matices ae used to pove a conjectue due to Kalin and Rinott. 1. Intoduction. If A = ((a^-)) is an n x n matix, the pemanent of A, denoted by pe A, is defined as n pea= ]T Y[aMi)' < Sn i=l whee Sn is the goup of pemutations of 1,2,..., n. The pobability density function of a multinomial distibution can be expessed in tems of the pemanent of a suitable matix. To begin with a simple example, suppose X denotes the numbe of heads esulting fom n tosses of a coin, with pobability of heads equal to p, 0 < p < 1, on a single toss. Then X has the Binomial distibution and its pobability density function is given by (1) P(X = x)= n!.,p* g""*, x = 0,l,...,n; x\(n x)\ whee P(X = x) denotes the pobability that X equals x, q = 1 p, and, as usual, P(X = x) is undestood to be zeo fo all values of x not specified in (1). The pobability in (1) can be expessed in tems of a pemanent as follows: P p «P^X = ^ = xw^.^ P a x times, x = 0,1,...,n. (n x) times, lq flj Received by the editos July 11, 1986 and, in evised fom, Decembe 15, Pesented at the Second Utah State Matix Theoy Confeence held in Logan, Utah, Januay 29-31, 1987, sponsoed by the Depatment of Mathematics, Utah State Univesity Mathematics Subject Classification (1985 Revision). Pimay 15A15, 60E15; Seconday 15A48, 94A17. Key wods and phases. Pemanents, multinomial distibution, conditionally negative definite matices Ameican Mathematical Society /88 $ $.25 pe page License o copyight estictions may apply to edistibution; see

2 468 R. B. BAPAT The expession in (2) admits genealizations. Fo example, suppose n diffeent coins ae tossed once and let X be the numbe of heads obtained. If p is the pobability of heads on a single toss of the ith coin and if ç, = 1 p, i = 1,2,..., n, then it can be veified that Pi Pn (3) P(X = x) = x\(n x)\ pe Pi Qi Pn <?n x times, X = 0, 1,...,71. (n x) times, qi Qn Moe geneally, instead of tossing n coins, it may be an expeiment of olling n dice, diffeently loaded, and if A denotes the numbe of times i spots ae obtained, i = 1,2,..., 6, the density function of (Xi,..., Xq) can be witten in tems of a pemanent. To make the concepts pecise, conside an expeiment which can esult in any of possible outcomes and suppose n tials of the expeiment ae pefomed. Let ttí3 be the pobability that the expeiment esults in the ith outcome at the jth tial, i 1,2,..., ; j = 1,2,..., n. Let n denote the x n matix ((ttí3 )), which, of couse, is column stochastic. Let Xi denote the numbe of times the z'th outcome is obtained in the n tials, i = l,2,...,, and let X = (Xi,..., X). In this setup we will say that X has the multipaamete multinomial distibution with the xn paamete matix n. Let 7* denote the ith ow of n, i = 1,2,...,. The density function of X can again be expessed in tems of a pemanent as -7T1 (4) P(X = t) = h\-t pe t G Kn, t times, whee Kn = \k (ki,...,k): fc, nonnegative integes, V~^fc = n >. It seems that the epesentation (4) is impotant in undestanding cetain popeties of the multinomial distibution. The pupose of this pape is to exploit (4) to settle a conjectue due to Kalin and Rinott [4]. 2. The poblem. Befoe stating the main poblem, we need some notation. Let H =\xgr: xt=o[. i=l License o copyight estictions may apply to edistibution; see

3 MULTINOMIAL PROBABILITIES 469 DEFINITION 1. A eal, symmetic x matix A is said to be conditionally negative definite (c.n.d.) if fo any x G H, Y2^2ai]XiX3 <o. =i j=i Now suppose X = (Xi,...,X) follows the multipaamete multinomial distibution with the x n paamete matix n = ((ttí3)). We assume thoughout that n is positive and that n > 2, > 2. Fix k G Kn-2 and let (O) Kij = [fci,..., Ki 1, ft» T 1, Ät+1)»"j li Kii = \Kl,.., Ki 1, fci + >, Ki+i,..., K), 1 S: t T. fcj T 1, Kj+1,, "V)) 1 <» # 3 <, Clealy, kij G Kn, i,j = 1,2,...,. Ou main esult is that ((logp(x = ki3))) is c.n.d. This appeas as Conjectue 2.1 in Kalin and Rinott [4], whee it has been confimed fo = 2,3 and fo any n. We efe to [4] fo a discussion concening the elevance of the poblem to multivaiate majoization and inequalities as well as fo cetain consequences of poving the conjectue. In paticula, Conjectue 2.2 of [4] is also veified once Conjectue 2.1 is established. It must be emaked that we have used a notation diffeent than that of [4] at vaious places in the pape. Also, we find it moe convenient to wok with conditionally negative definite matices athe than conditionally positive definite matices. 3. Pemanents of positive matices. In this section we give cetain esults on pemanents that will be used. Let A be a eal matix of ode (n 2) x n, n > 2. Let e3 denote the unit ow vecto (0,..., 0,1,0,..., 0), whee the 1 occus at the jth place, j = 1,2,..., n. Define an n x n matix A as follows. The (i,j)th enty of A is the pemanent of the augmented matix i,3 = 1,2,..., n. Note that the diagonal enties of À ae all zeo and À is symmetic. if n = 2, then A is absent and A = Of couse The following theoem is a deep esult in the theoy of pemanents. It was oiginally poved by Alexandoff [1] in a moe geneal fom. It seves as a cucial step in the poof of the well-known van de Waeden conjectue due to Egoychev [2, 10] and Falikman [3]. Fo a poof of the esult, see Theoem 2.8 of [10]. Again, bewae of diffeences in notation. THEOREM 2. If A is a positive (n 2)xn matix, n > 2, then the matix A is nonsingula and has exactly one, simple, positive eigenvalue. The following esult will be deduced fom Theoem 2. License o copyight estictions may apply to edistibution; see

4 470 R. B. BAPAT THEOREM 3. Let A be a positive (n 2) x n matix, n > 2, and let x1,..., be ow vectos in Rn. Define the x matix B = ((bi3)) as bi3 = pe A xi x>»', = 1,2,. Then B has at most one, simple, positive eigenvalue. PROOF. The pemanent [7, p. 16]. Expand admits a Laplace expansion in tems of a set of ows pe x in tems of the last two ows and obseve that (6) 6,,-iA^), i,j = 1,2,...,. Let X be the xn matix whose th ow is xx, i 1,2,...,. Then it follows fom (6) that B = XAX'. By Theoem 2, A has exactly one positive eigenvalue and hence (essentially) by Sylveste's law, B has at most one positive eigenvalue. This completes the poof. 4. Conditionally negative definite matices. The definition of a c.n.d. matix was given in 2. The following esult will be useful (see, fo example, Pathasaathy and Schmidt [8, p. 3]). LEMMA 4. A eal, symmetic matix A is c.n.d. if and only if fo each a > 0, the matix ((e~aaij)) is positive semidefinite. Recall that a function F: (0, oo)» R is said to be completely monotonie if it is in C (0,oo) and (-l)fcfw(x) > 0, x G (0,oo), k = 0,1,2,..., whee, by definition, F^ = F. The following esult appeas in a ecent pape by Micchelli [6]. A poof is included fo completeness. LEMMA 5. Let A be a c.n.d. nxn matix with positive enties and let F: (0, oo) R be completely monotonie. Then the matix ((F(üí3))) is positive semidefinite. PROOF. By a well-known theoem of Benstein (see, fo example, [11, p. 160]), F admits the epesentation oo -ta F(t) = / Jo dp(o), t > 0, whee dp(o) is a Boel measue on (0, oo). The esult follows by Lemma 4. Now we have the following. LEMMA 6. Let A be a symmetic, positive x matix with exactly one, simple, positive eigenvalue. Then, fo any x G H,»,j=i n c ^ i- License o copyight estictions may apply to edistibution; see

5 MULTINOMIAL PROBABILITIES 471 PROOF. It is known that if A is a symmetic, positive x matix, then thee exists a positive vecto z such that the matix B = ((bi3)) = ((üí3zíz3)) is doubly stochastic (see, fo example, [5, 9]). Clealy, if A satisfies the hypothesis of the lemma, then B also has only one positive eigenvalue by Sylveste's law, and, futhemoe, il b*?> = [I (al3ziz3y*> = n a*p fo any x G H.»,i=i Thus, we may assume, without loss of geneality, that A is doubly stochastic, so that the vecto (1,..., 1) is an eigenvecto of A coesponding to the eigenvalue 1. Since A has only one positive eigenvalue, any vecto x G H lies in the span of eigenvectos of A coesponding to only nonpositive eigenvalues and, hence, ^2^a,i3xtx3 < 0. i=ij=i Thus, A is c.n.d. The function F(t) = t~a, t > 0, is completely monotonie fo any a > 0 and by Lemma 5, ((aza)) is positive semidefinite. Hence, fo any a > 0, ((e_qlog"' >)) is positive semidefinite and by Lemma 4, ((logaij)) is c.n.d. This completes the poof. 5. The main esult. We ae now in a position to pove the following statement, conjectued by Kalin and Rinott [4]. THEOREM 7. Let X = (Xi,...,XT) have the multipaamete multinomial distibution with the x n paamete matix Tí. Let k G Kn-2, let kij be defined as in (5), and let m,i3 = P(X = kij), i,j = 1,2,...,. Then the matix ((logm,j)) is c.n.d. PROOF. We have to pove that fo any x G H, (7) fl m*p < 1.»,i=i Let A be the (n 2) x n matix fomed by taking ki copies of 7\ the ith ow of n, i 1,2,..., n. Define the x matix B = ((hj)) as bij = pe A TT3 i,j = 1,2,...,. By Theoem 3, B has at most one positive eigenvalue, and since B is a positive matix, it must have exactly one positive eigenvalue. So by Lemma 6, (8) Yl h*p <1 fo any x G H. i,j=l License o copyight estictions may apply to edistibution; see

6 472 R. B. BAPAT By the elationship (4), (9) mi3 = P(X = kl3) = (ki\,...,kl)(ki + l)(k3 + l) bu I (ki\---k\)(kl + l)(ki+2y \<i j<, Ki<. Let D be the x diagonal matix with its ith diagonal enty equal to (fc +1)-1, t = 1,2,...,, and let M C=k^kl.DBD- It follows fom (8) and (10) that fo any x G H, (il) n c*p < i. *ij=l Fom (9) and (10), (i2) m«={(;i:+i )/(ki + 2))di, 1 < «7e j <, l<i<, Since (ki + l)/(/c + 2) < 1, i = 1,2,...,, it follows fom (12) and (11) that H m p i,j=l and the poof is complete. < } [ c*p < 1 fo any x G HT, i,j=l Refeences 1. A. D. Alexandoff, Zu théoie de gemischten Volumina von konvexen köpen. IV, Mat. Sb. 45 (1938), no. 3, (Russian; Geman summay). 2. G. P. Egoychev, The solution of van de Waeden's poblem o pemanents, Adv. in Math. 42 (1981), D. I. Falikman, A poof o van de Waeden's conjectue on the pemanent of a doubly stochastic matix, Mat. Zametki 29 (1981), (Russian) 4. S. Kalin and Y. Rinott, Entopy inequalities o classes o pobability distibutions. II, The multivaiate case, Adv. in Appl. Pobab. 13 (1981), M. Macus and M. Newman, The pemanent o a symmetic matix, Notices Ame. Math. Soc. 8 (1981), C. A. Micchelli, Intepolation o scatteed data: Distance matices and conditionally positive definite matices, Const. Appox. 2 (1986), H. Mine, Pemanents, Encyclopedia of Mathematics and its Applications, Vol. 6, Addison- Wesley, Reading, Mass., K. R. Pathasaathy and K. Schmidt, Positive definite kenels, continuous tenso poducts and cental limit theoems o pobability theoy, Spinge-Velag, Belin, R. Sinkhon, A elationship between abitay positive matices and doubly stochastic matices, Ann. Math. Statist. 35 (1964), J. H. van Lint, Notes on Egoitsjev's poof of the van de Waeden conjectue, Linea Algeba Appl. 39 (1981), D. V. Widde, The Laplace tans om, Pinceton Univ. Pess, Pinceton, N.J., indian statistical institute, delhi cente 7, s. j. s. sansanwal mac new Delhi , India License o copyight estictions may apply to edistibution; see