UPPER AND LOWER BOUNDS FOR THE TAILS OF THE DISTRIBUTION OF THE CONDITION NUMBER OF A GAUSSIAN MATRIX

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1 UPPER AND LOWER BOUNDS FOR THE TAILS OF THE DISTRIBUTION OF THE CONDITION NUMBER OF A GAUSSIAN MATRIX JEAN-MARC AZAïS AND MARIO WSCHEBOR Abstract. Let A be an m m real random matrix with i.i.d. standard Gaussian entries. We prove that there exist universal positive constants c and C such that the tail of the probability c distribution of the condition number κ(a) satisfies the inequalities x < P{κ(A) > mx} < C x for every x >. The proof requires a new estimation of the joint density of the largest and the smallest eigenvalues of A T A which follows from a formula for the expectation of the number of zeros of a certain random field defined on a smooth manifold. by Key words. Random matrices, Condition Number, Eigenvalue Distribution, Rice Formulae. AMS subject classifications. 5A 6G5. Introduction and main result. Let A be an m m real matrix and denote A = sup Ax x = its Euclidean operator norm. x denotes Euclidean norm of x in R m. If A is nonsingular, its condition number κ(a) is defined by κ(a) = A A (von Neumann and Goldstine [8], Turing [7]). The role of κ(a) in a variety of numerical analysis problems is well-established (see for example Wilkinson [], Smale [4], Higham [9], Demmel [6]). The purpose of the present paper is to prove the following Theorem.. Assume that A = ((a ij )) i,j=,...,m, m 3, and that the a ij s are i.i.d. Gaussian standard random variables. Then, there exist universal positive constants c, C such that for x > : c x < P{κ(A) > mx} < C x. (.) Remarks on the statement of Theorem... It is well-known that as m tends to infinity, the distribution of the random variable κ(a)/m converges to a certain distribution (this follows easily, for example, from Edelman [7] ). The interest of (.) lies in the fact that it holds true for all m 3 and x >. We will see below that c =.3, C = 5.6 satisfy (.) for every m = 3, 4,... and x >. Using the same methods one can obtain more precise upper and lower bounds for each m but we will not detail these calculations here. The simulation study of Section 3 suggests that c must be smaller than.87 and C larger than.8. Laboratoire de Statistique et Probabilités. UMR-CNRS C5583 Université Paul Sabatier. 8, route de Narbonne. 36 Toulouse Cedex 4. France. azais@cict.fr Centro de Matemática. Facultad de Ciencias. Universidad de la República. Calle Igua Montevideo. Uruguay..wschebor@cmat.edu.uy

2 J-M AZAÏS and M. WSCHEBOR 3. In Sankar, Spielman and Teng [3] it was conjectured that P{κ(A) > x} = O( m σx ) when the a ij s are independent Gaussian random variables having a common variance σ and sup i,j E (a ij ). The upper bound part of (.) implies that this conjecture holds true in the centered case. The lower bound shows that, up to a constant factor, this is the exact order of the behavior of the tail of the probability distribution of κ(a). See Wschebor [] for the non centered case. 4. This Theorem, and related ones, can be considered as results on the Wishart matrix A T A (A T denotes the transpose of A). Introducing some minor changes, it is possible to use the same methods to study the condition number of A T A for rectangular n m matrices A having i.i.d. Gaussian standard entries, n > m. This will be considered elsewhere. Some examples of related results on the random variable κ(a) are the following. Theorem. ([7]). Under the same hypothesis of Theorem., one has: E (log κ(a)) = log m + C + ε m where C is a known constant (C.537) and ε m as m +. Theorem.3 ([4]). Let A = ((a ij )) i,j=,...,m and assume that the a ij s are independent Gaussian random variables with a common variance σ and m ij = E(a ij ). Denote by M = ((m ij )) i,j=,...,m the non-random matrix of expectations. Then, [ ] M E (log κ(a)) log m + log σ m C, where C is a known constant. Next, we introduce some notation. Given A, an m m real matrix, we denote by λ,..., λ m, λ... λ m the eigenvalues of A T A. If X : S m R is the quadratic polynomial X(x) = x T A T Ax, then: λ m = A = max x S m X(x) in case λ >, λ = = min A x S m X(x). It follows that κ(a) = ( λm when λ >. We put κ(a) = + if λ =. Note also that κ(a) and κ(ra) = κ(a) for any real r, r. There is an important difference between the proof of Theorem. and those of the two other theorems mentioned above. In the latter cases, one puts λ ) log κ(a) = log λ m log λ and if one takes expectations, the joint distribution of the random variables λ m, λ does not play any role; the proof only uses the individual distributions of λ m and λ. On the contrary, the proof below of Theorem. depends essentially on the joint

3 TAILS OF THE DISTRIBUTION OF THE CONDITION NUMBER 3 distribution of the pair (λ m, λ ). A general formula for the joint density of λ,..., λ m is well-known since a long time (see for example Wilks [], Wigner [9], Krishnaiah and Chang [], Kendall, Stuart and Ord [], and references therein) but it seems to be difficult to adapt this to our present requirements. In fact, we will use a different approach, based upon the expected value of the number of zeros of a random field parameterized on a smooth manifold. We have also applied this technique to give a new proof of the known result of Lemma, a lower bound for P{λ < a}. One can ask if Theorem. follows from the well-known exponential bounds for the concentration of the distribution of λ m together with known bounds for the distribution of λ (see for example Szarek [5], Davidson and Szarek [5] and Ledoux [] for these type of inequalities). More precisely, consider the upper-bound in Theorem.. For ε > one has: { λm P { κ(a) > mx } = P > m x } λ P { λ m > (4 + ε)m } { + P λ m (4 + ε)m, λ m > m x } λ P { λ m > (4 + ε)m } { (4 + ε) } + P λ < mx C exp [ C mε ] 4 + ε + C 3, (.) x where C, C, C 3 are positive constants. From (.), making an adequate ( choice ) of α ε one can get an upper bound for P{κ(A) > mx} of the form (const) log x x m for some α > and x large enough. However, this kind of argument does not lead to the precise order given by our Theorem.. On the other hand, using known results for the distribution of other functions of the spectrum (for example (λ λ m )/λ as in Edelman [8]) one can get upper and lower bounds for the tails of the distribution of κ(a) which again do not reach the precise behavior (const)/x.. Proof of Theorem.. It is easy to see that, almost surely, the eigenvalues of A T A are pairwise different. We introduce the following additional notations: <.,. > is usual scalar product in R m and {e,..., e m } the canonical basis. I k denotes the k k identity matrix. B = A T A = ((b ij )) i,j=,...,m For s in R m, π s : R m R m denotes the orthogonal projection onto {s}, the orthogonal complement of s in R m. M (resp. M ) means that the symmetric matrix M is positive definite (resp. negative definite). If ξ is a random vector, p ξ (.) is the density of its distribution whenever it exists. For a differentiable function F defined on a smooth manifold M embedded in some Euclidean space, F (s) and F (s) are the first and the second derivative of F that we will represent, in each case, with respect to an appropriate orthonormal basis of the tangent space. Instead of (.) we prove the equivalent statement: for x > m, cm x < P{κ(A) > x} < Cm x. (.)

4 4 J-M AZAÏS and M. WSCHEBOR We break the proof into several steps. Our main task is to estimate the joint density of the pair (λ m, λ ); this will be done in Step 4. Step. For a, b R, a > b, one has almost surely: {λ m (a, a + da), λ (b, b + db)} { s, t S = m, < s, t >=, X(s) (a, a + da), X(t) (b, b + db), π s (Bs) =, π t (Bt) =, X (s), X (t) An instant reflection shows that almost surely the number } (.) N a,b,da,db of pairs (s, t) belonging to the right-hand side of (.) is equal to or to 4, so that: P{λ m (a, a + da), λ (b, b + db)} = 4 E(N a,b,da,db). (.3) Step. In this step we will give a bound for E (N a,b,da,db ) using what we call a Rice-type formula (see Azaïs and Wschebor [3], for some related problems and general tools). Let V = { (s, t) : s, t S m, < s, t >= }. V is a C -differentiable manifold without boundary, embedded in R m, dim(v ) = m 3. We will denote by τ = (s, t) a generic point in V and by σ V (dτ) the geometric measure on V. It is easy to see that σ V (V ) = σ m.σ m where σ m denotes the surface area of S m R m, that is σ m = πm/ Γ(m/). On V we define the random field Y : V R m by means of Y (s, t) = ( πs (Bs) π t (Bt) ). For τ = (s, t) a given point in V, we have that Y (τ) {(t, s)} { {s} {t} } = W τ for any value of the matrix B, where {(t, s)} is the orthogonal complement of the point (t, s) in R m. In fact, (t, s) {s} {t} and < Y (s, t), (t, s) > R m=< π s (Bs), t > < π t (Bt), s > =< Bs < s, Bs > s, t > < Bt < t, Bt > t, s >= since < s, t >= and B is symmetric. Notice that dim(w τ ) = m 3. We also set (τ) = [ [ ]] det (Y (τ)) T Y (τ)

5 TAILS OF THE DISTRIBUTION OF THE CONDITION NUMBER 5 For τ = (s, t) V, F τ denotes the event N = {τ : τ V, Y (τ) = }. F τ = {X(s) (a, a + da), X(t) (b, b + db), X (s), X (t) }, and p Y (τ) (.) is the density of the random vector Y (τ) in the (m 3)-dimensional subspace W τ of R m. Assume that is not a critical value of Y, that is, if Y (τ) = then (τ). This holds true with probability. By compactness of V, this implies N <. Assume that N and denote by τ,..., τ N the roots of the equation Y (τ) =. Because of the implicit function theorem, if δ > is small enough, one can find in V open neighborhoods U,..., U N of the points τ,..., τ N respectively, so that: Y is a diffeomorphism between U j and Y (V ) B m (, δ) (B m (, δ) is the Euclidean ball of radius δ centered at the origin, in R m ). U,..., U N are pairwise disjoint. If τ / N j= U j, then Y (τ) / B m (, δ). Using the change of variable formula, it follows that: V (τ) { Y (τ)<δ } σ V (dτ) = N j= U j (τ) σ V (dτ) = N µ (Y (U j )), (.4) where µ (Y (U j )) denotes the - (m 3)-dimensional - geometric measure of Y (U j ). As δ, µ (Y (U j )) B m 3 (δ) where B m 3 (δ) is the (m 3)-dimensional Lebesgue measure of a ball of radius δ in R m 3. It follows from (.4) that, almost surely, N = lim δ B m 3 (δ) V j= (τ)i { Y (τ)<δ } σ V (dτ). In exactly the same way, one can prove that N a,b,da,db = lim (τ)i Fτ I δ B m 3 (δ) { Y (τ) <δ} σ V (dτ). Applying Fatou s Lemma and Fubini s Theorem: E(N a,b,da,db ) lim inf E ( ) (τ) I Fτ I { Y (τ)<δ } σv (dτ) δ B m 3 (δ) V = lim inf σ V (dτ) E ( (τ)i Fτ /Y (τ) = y) p Y (τ) (y) δ V B m,δ,τ = E ( (τ)i Fτ /Y (τ) = ) p Y (τ) () σ V (dτ), V V dy B m 3 (δ) where B m,δ,τ = B m (, δ) W τ. The validity of the last passage to the limit will become clear below, since it will follow from the calculations we will perform, that the integrand in the inner integral is a continuous function of the pair (τ, y). Hence: E (N a,b,da,db ) a+da a dx b+db b dy V E ( (s, t)i {X (s),x (t) }/C s,t,x,y ).p X(s),X(t),Y (s,t) (x, y, ) σ V (d(s, t)), (.5)

6 6 J-M AZAÏS and M. WSCHEBOR where C s,t,x,y is the condition: {X(s) = x, X(t) = y, Y (s, t) = }. The invariance of the law of A with respect to isometries of R m implies that the integrand in (.5) does not depend on (s, t) V. Hence, we have proved that the joint law of λ m and λ has a density g(a, b), a > b, and g(a, b) 4 σ m σ m E ( ) (e, e )I {X (e ),X (e ) }/C e,e,a,b p X(e),X(e ),Y (e,e )(a, b, ). (.6) In fact, using the method of Azaïs and Wschebor [3], it could be proved that (.6) is an equality, but we do not need such a precise result here. Step 3. Next, we compute the ingredients in the right-hand member of (.6). We take as orthonormal basis for the subspace W (e,e ): We have: {(e 3, ),..., (e m, ), (, e 3 ),..., (, e m ), X(e ) = b X(e ) = b X (e ) = B b I m X (e ) = B b I m, (e, e )} = L. where B (resp. B ) is the (m ) (m ) matrix obtained by suppressing the first (resp. the second) row and column in B. Y (e, e ) = (, b, b 3,..., b m, b,, b 3,..., b m, b ) T so that, it has the expression in the orthonormal basis L : Y (e, e ) = m ( bi (e i, ) + b i (, e i ) ) + ( b (e, e ) ). i=3 It follows that the joint density of X(e ), X(e ), Y (e, e ) appearing in (.6) in the space R R W (e,e ) is the joint density of the r.v. s b, b, b, b 3,..., b m, b 3,..., b m at the point (a, b,,..., ). To compute this density, first compute the joint density q of b 3,..., b m, b 3,..., b m, given a, a, where a j denotes the j-th column of A which is Gaussian standard in R m. q is the normal density in R (m ), centered with variance matrix ( ) a I m < a, a > I m < a, a > I m a. I m Set a j = a j, j =,. a j

7 TAILS OF THE DISTRIBUTION OF THE CONDITION NUMBER 7 The density of the triplet (b, b, b ) = ( a, a, a a < a, a >) at the point (a, b, ) can be computed as follows. Since < a, a > and ( a, a ) are independent, the density of the triplet at (a, b, ) is equal to: χ m(a)χ m(b)(ab) / p <a,a > () where χ m (.) denotes the χ density with m degrees of freedom. Let ξ = (ξ,..., ξ m ) T be Gaussian standard in R m. Clearly, < a, a > has the same distribution as ξ ξ, because of the invariance under rotations. t P{ < a, a > t} = t P{ ξ χ m t t } = t P{F,m t (m ) t } = t t (m ) t f,m (x)dx, where χ m = ξ ξ m and F,m has the Fisher distribution with (, m ) degrees of freedom and density f,m. Letting t, we obtain p <a,a > () = Γ(m/) π Γ ( (m )/ ). Summing up, the density in (.6) is equal to : (π) m π Γ(m/)Γ ( (m )/ ) m exp ( a + b ). (.7) ab We now consider the conditional expectation in (.6). First, observe that the (m 3)- dimensional tangent space to V at the point (s, t) is parallel to the orthogonal complement in R m R m of the triplet of vectors (s, ); (, t); (t, s). This is immediate from the definition of V. To compute the associated matrix for Y (e, e ) take the set {(e 3, ),..., (e m, ), (, e 3 ),..., (, e m ), (e, e )} = L. as orthonormal basis in the tangent space and the canonical basis in R m. A direct calculation gives : v T,m b w T,m ( b + b ) Y B (e, e ) = b I m m,m w,m w T ( b + b ),,m v T b m,m B b I m v where v T = (b 3,..., b m ), w T = (b 3,..., b m ), i,j is a null matrix with i rows and j columns and B is obtained from B by suppressing the first and second rows and

8 8 J-M AZAÏS and M. WSCHEBOR columns. The columns represent the derivatives in the directions of L at the point (e, e ). The first m rows correspond to the components of π s (Bs), the last m ones to those of π t (Bt). Thus, under the condition C e,e,a,b that is used in (.6), Y (e, e ) = and [ det [( Y (e, e ) ) T Y (e, e ) ]],m,m,m,m (b a) B ai m m,m m,,m,m (b a),m,m m,m B bi m m, = det(b ai m ) det(b bi m ) (a b). Step 4. Note that B ai m B ai m, and similarly, B bi m B bi m, and that for a > b, under C e,e,a,b, there is equivalence in these relations. It is also clear that, since B one has det(b ai m ) I B ai m a m and it follows that the conditional expectation in (.6) is bounded by: a m E ( det(b bi m ) I B bi m /C ), (.8) where C is the condition {b = a, b = b, b =, b i = b i = (i = 3,..., m)}. To compute the conditional expectation in (.8) we further condition on the value of the random vectors a and a. Since unconditionally a 3,..., a m are i.i.d. standard Gaussian vectors in R m, under this new conditioning, their joint law becomes the law of i.i.d. standard Gaussian vectors in R m and independent of the condition. That is, (.8) is equal to a m E ( det(m bi m ) I M bim ), (.9) where M is an (m ) (m ) random matrix with entries M ij =< v i, v j >, (i, j =,..., m ) and the vectors v,...v m are i.i.d. Gaussian standard in R m. The expression in (.9) is bounded by a m E ( det(m) ) = a m (m )!, The last equality is contained in the following lemma, which is well-known, see for example Edelman [7]. Lemma.. Let ξ,..., ξ m be i.i.d. random vectors in R p, p m, their common distribution being Gaussian centered with variance I p. Denote W m,p the matrix and its characteristic polynomial. Then W m,p = ((< ξ i, ξ j >)) i,j=,...,m. D(λ) = det (W m,p λi m )

9 TAILS OF THE DISTRIBUTION OF THE CONDITION NUMBER 9 (i) E (det (W m,p )) = p(p )...(p m + ) (.) (ii) E (D(λ)) = m ( ) m ( ) k p! k (p m + k)! λk (.) k= Returning to the proof of the Theorem and summing up this part, after replacing in (.6), we get g(a, b) C m exp ( (a + b)/ ) ab a m, (.) where C m = 4(m )!. Step 5. Now we prove the upper-bound part in (.). One has, for x > P{κ(A) > x} = P{ λ m λ > x } P{λ < L m x } + P{λ m λ > x, λ L m }, (.3) x where L is a positive number to be chosen later on. For the first term in (.3), we use Proposition 9 in Cuesta-Albertos and Wschebor [4] which is a slight modification of Theorem 3. in Sankar, Spielman and Teng [3]: Here, P{λ < L m x } = P{ A > x L m } C (m) Lm x. C (m) = ( ) [ { sup cp t π m > <c<m (m )c}] C (+ ).3473, m c where t m is a random variable having Student s distribution with m degrees of freedom. For the second term in (.3), with P{ λ m λ > x, λ L m + + x } = db g(a, b)da G m (x ) L mx bx G m (y) = C m + + db L my by exp ( (a + b)/ ) a m da, ab using (.). We have: [ + G m(y) = C m exp( b/) b exp ( (by)/ ) (by) m 3/ db+ L my L my + L m exp ( (a + L m y )) a m 3/ L m y da ] (.4)

10 J-M AZAÏS and M. WSCHEBOR which implies Put I m (a) = + G m (y) C my m 3/ exp ( b( + y) ) b m db L my ( ) m = y 3/ y + m e z z m dz 4(m )! + y L m y (+y) + a y 3/ 4(m )! m + L m e z z m dz. e z z m dz. Integrating by parts: I m (a) = e a[ a m + (m )a m + (m )(m )a m (m )! ], so that for a >.5m I m (a) 5 3 e a a m. If L > 5 we obtain the bound G m(y) D m y 3/ with D m = 5 m m ( ) 6 (m )! L(m ) exp L m. We now apply Stirling s formula (Abramowitz and Stegun [], 6..38) i.e. for all x > to get Γ(x + ) exp( ( x ) x x ) πx Γ(x + ), e D m 5 πl m exp( m L 4log(L) m ) 5 πl m, if we choose for L the only root larger than of the equation L 4log(L) = (check that L.345). To finish, G m (y) = + y G m (t)dt < D m + y dt t 3/ = D my. Replacing y by x and performing the numerical evaluations, the upper bound in (.) follows and we get for the constant C the value 5.6. Step 6. We consider now the lower bound in (.). For γ > and x >, we have : P{κ(A) > x} = P { λ m λ > x } P { λ m λ = P { λ < γ m x > x, λ < γ m x } } P { λ m λ x, λ < γ m x }. (.5) A lower bound for the first term in the right-hand member of (.5) is obtained using the following inequality, that we state as a separate lemma. In fact, this result is

11 TAILS OF THE DISTRIBUTION OF THE CONDITION NUMBER known, see for example Szarek [5] Th..., where it is proved without giving an explicit value for the constant. See also Edelman [7], Cor. 3., for a related result. Lemma.. If < a < /m, then P{λ < a} β am, where we can choose β = ( 3) 3/e /3. Proof. Define the index i X (t) of a critical point t S m of the function X as the number of negative eigenvalues of X (t). For each a > put N i (a) = {t S m : X(t) = t T Bt < a, X (t) =, i X (t) = i}, for i =,,..., m. One easily checks that if the eigenvalues of B are λ,..., λ m, < λ <... < λ m, then: Now consider if a λ then N i (a) = if λ i < a λ i+ then N k (a) = for i =,,..., m for some i =,,..., m for k =,..., i N k (a) = if λ m < a then N i (a) = m M(a) = ( ) i N i (a). i= for k = i,..., m for i =,,..., m. M(a) is the Euler characteristic of the set S = {t S m : X(t) < a}, see Adler []. It follows from the relations above that if N (a) =, then N i (a) = for i =,..., m, hence M(a) =, if N (a) =, then M(a) = or, so that in any case Hence, M(a) N (a). P{λ < a} = P{N (a) = } = E( N (a) ) E( M(a) ). (.6) The expectation of M(a) can be written using the following Rice-type formula, Azaïs and Wschebor [3] or Taylor and Adler [6] E ( M(a) ) = = a a dy E [ det ( X (t) ) /X(t) = y, X (t) = ] p X(t),X (t)(y, )σ m (dt) S m σ m (S m )E [ det ( X (e ) ) /X(e ) = y, X (e ) = ] p X(e),X (e )(y, )dy.

12 J-M AZAÏS and M. WSCHEBOR where we have used again invariance under isometries. Applying a similar Gaussian regression - as we did in Step 4 to get rid of the conditioning - we obtain: E ( M(a) ) = a E [ det ( )] π Q yi m m Γ ( m)exp( y/) dy (.7) y where Q is an (m ) (m ) random matrix with entry i, j equal to (< v i, v j >) and v,..., v m are i.i.d. Gaussian standard in R m. We now use part (ii) of Lemma. : E [ det ( m )] Q yi m = (m )! k= ( m k ) ( y) k k! (.8) Under condition < a < m, since < y < a, as k increases the terms of the sum in the right-hand member of (.8) have decreasing absolute value, so that: E [ det ( Q yi m )] (m )![ (m )y]. Substituting into the right-hand member of (.7), we get : E [ M(a) ] π (m )! m Γ (m/) J m(a), where, using again < a < m : J m (a) = a ( )exp( y/) a (m )y dy y ( (m )y ) y ( y/)dy 4 3 a, by an elementary computation. Going back to (.7), applying Stirling s formula and remarking that ( + /n ) n+ e we get P{λ < a} ( ) 3/e /3 am. 3 This proves the lemma. End of the proof of Theorem. Using Lemma., the first term in the right hand member of (.5) is bounded below by βγ m x. To obtain a bound for the second term, we use again our upper bound (.) on the joint density g(a, b), as follows: { λm P x, λ < γ m λ γ m x C m bx db b } x = γ m x exp ( (a + b)/ ) ab 4(m )! bx db g(a, b)da b a m da C m γ m x x x 3 γ m m m b(x )b (bx ) m 3 db 8 π em γ m m x, (.9)

13 TAILS OF THE DISTRIBUTION OF THE CONDITION NUMBER 3 Table 3. Values of the estimations P{κ(A) > mx} for x =,, 3, 5,, 5, 3, 5, and m = 3, 5,, 3,, 3, 5 by Monte-Carlo method over 4 simulations. Value of x Probability Lower b.:.3/x Upper b.: 5.6/x m = m= m= m= m= m= m = on applying Stirling s formula. Choosing now γ = /e, we see that the hypothesis of Lemma. is satisfied and also { λm P x, λ < γ m } λ x 8 m π e 3 x. Replacing into (.5), we obtain the lower bound in (.) with c = ( ) 3/e 4/3 3 8 π e Monte-Carlo experiment. To study the tail of the distribution of the condition number of Gaussian matrices of various size, we used the following Matlab functions normrnd to simulate normal variables cond to compute the Condition number of matrix A. Results over 4 simulation using Matlab are given in Table 3.. This table suggests that the constants c and C should take values smaller than.87 and bigger than.8 respectively. Acknowledgments. The authors thank Professors G. Letac and F. Cucker for valuable discussions. REFERENCES [] Milton Abramowitz and Irene A. Stegun, editors. Handbook of mathematical functions with formulas, graphs, and mathematical tables. A Wiley-Interscience Publication. John Wiley & Sons Inc., New York, 984. Reprint of the 97 edition, Selected Government Publications. [] Robert J. Adler. The geometry of random fields. John Wiley & Sons Ltd., Chichester, 98. Wiley Series in Probability and Mathematical Statistics. [3] J-M. Azaïs and M. Wschebor. On the distribution of the maximum of a gaussian field with d parameters. Annals Appl. Prob., to appear, 4. [4] J. Cuesta-Albertos and M. Wschebor. Condition numbers and the extrema of random fields. In proc. IV Ascona Seminar. Birkaüser, Boston, to appear, 4. [5] Kenneth R. Davidson and Stanislaw J. Szarek. Local operator theory, random matrices and Banach spaces. In Handbook of the geometry of Banach spaces, Vol. I, pages North-Holland, Amsterdam,.

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