Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

On Berry-Esséen rates,a law of the iterated logarithm and an invariance principle for the proportion of the sample below the sample mean

Let F_n(x) be the empirical distribution function based on n independent random variables X₁,…,X_n from a common distribution function F(x), and let $\text{X}\text{= Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{X}{}_{\mathrm{i}}\text{n}$ be the sample mean. We derive the rate of convergence of $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$ to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$.     

A note on the proportion of singular (0, 1)-matrices in a class associated with fractional 2n factorial designs

Let the set T = {(x₁, x₂,…, x_n): x_i = 0, 1}. Since the elements of T can be seen as binary representations of integers, we order them with their corresponding integer values. Let $\text{Γ}{}^1$ be the set of (n + 1) × (n + 1) matrices of the form [1 … D], where the n + 1 rows of D are distinct ordered elements of T. We show that the proportion of singular matrices in $\text{Γ}{}^1$ approaches 0 as n → ∞.     

The asymptotic equivalence of Bayes and maximum likelihood estimation

Let (X, $\text{A}$) be a measurable space, Θ ? $\text{R}$ an open interval and P_Ω ∥ $\text{A}$, Ω ? Θ, a family of probability measures fulfilling certain regularity conditions. Let $\text{Ω}{}_{\mathrm{n}}$ be the maximum likelihood estimate for the sample size n. Let λ be a prior distribution on Θ and let $\text{R}{}_{\mathrm{<mtext>n,</mtext><mtext>x</mtext>}}$ be the posterior distribution for the sample size n given $\text{x}\text{? X}{}^{\mathrm{n}}$. $\text{L:}\text{Θ}\text{× Θ →}\text{R}$ denotes a loss function fulfilling certain regularity conditions and T_n denotes the Bayes estimate relative to λ and L for the sample size n. It is proved that for every compact K ? Θ there exists c_K ≥ 0 such that

This theorem improves results of Bickel and Yahav [3], and Ibragimov and Has'minskii [4], as far as the speed of convergence is concerned.     

Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

A generalization of Turán's theorem to directed graphs

We consider an extremal problem for directed graphs which is closely related to Turán's theorem giving the maximum number of edges in a graph on n vertices which does not contain a complete subgraph on m vertices. For an integer n?2, let T_n denote the transitive tournament with vertex set X_n={1,2,3,…,n} and edge set {(i,j):1?i<j?n}. A subgraph H of T_n is said to be m-locally unipathic when the restriction of H to each m element subset of X_n consisting of m consecutive integers is unipathic. We show that the maximum number of edges in a m-locally unipathic subgraph of T_n is ($\text{q}\text{2}\text{)(m?1)}{}^2\text{+q(m?1)r+?}\text{1}\text{4}\text{r}{}^2\text{?}$ where n= q(m?1+r and $\text{?}\text{1}\text{2}\text{(m?1)??r<?}\text{3}\text{2}\text{(m?1)?}$. As is the case with Turán's theorem, the extremal graphs for our problem are complete multipartite graphs. Unlike Turán's theorem, the part sizes will not be uniform. The proof of our principal theorem rests on a combinatorial theory originally developed to investigate the rank of partially ordered sets.     

Inequalities concerning numbers of subsets of a finite set

Let S(n) denote the set of subsets of an n-element set. For an element x of S(n), let Γx and Px denote, respectively, all (|x| ?1)-element subsets of x and all (|x| + 1)-element supersets of x in S(n). Several inequalities involving Γ and P are given. As an application, an algorithm for finding an x-element antichain $\text{X}{}^1$ in S(n) satisfying $\text{| YX}{}^1\text{| ? | YX |}$ for all x-element antichains X in S(n) is developed, where YX is the set of all elements of S(n) contained in an element of X. This extends a result of Kleitman [9] who solved the problem in case x is a binomial coefficient.     

Rôle de l'espace de Besov B∞−1,∞ dans le contrôle de l'explosion éventuelle en temps fini des solutions régulières des équations de Navier–Stokes

Let $\text{u∈C([0,T}{}^1\text{[;L}{}^{\mathrm{n}}\text{(}\text{R}{}^{\mathrm{n}}\text{)}{}^{\mathrm{n}}\text{)}$ be a maximal solution of the Navier–Stokes equations. We prove that u is C^∞ on $\text{]0,T}{}^1\text{[×}\text{R}{}^{\mathrm{n}}$ and there exists a constant $\text{ε}{}_1\text{>0}$, which depends only on n, such that if $\text{T}{}^1$ is finite then, for all $\text{ω∈S(}\text{R}{}^{\mathrm{n}}\text{)}{}^{\mathrm{n}}\text{,}$ we have To cite this article: R. May, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

On the maximum cardinality of a consistent set of arcs in a random tournament

For any tournament T on n vertices, let h(T) denote the maximum number of edges in the intersection of T with a transitive tournament on the same vertex set. Sharpening a previous result of Spencer, it is proved that, if T_n denotes the random tournament on n vertices, then, $\text{P}$$\text{(h(T}{}_{\mathrm{n}}\text{) ≤}\text{1}\text{2}\text{(}{}_2{}^{\mathrm{n}}\text{) + 1.73n}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{) → 1}$ as n → ∞.     

Petites valeurs de la fonction d'Euler

Let φ be the Euler's function. A question of Rosser and Schoenfeld is answered, showing that there exists infinitely many n such that $\text{n}\text{φ(n)}\text{> e}{}^{\mathrm{y}}\text{log log}\text{n}$, where γ is the Euler's constant. More precisely, if N_k is the product of the first k primes, it is proved that, under the Riemann's hypothesis, $\text{N}{}_{\mathrm{k}}\text{φ(N}{}_{\mathrm{k}}\text{)}\text{> e}{}^{\mathrm{y}}\text{log log}\text{N}{}_{\mathrm{k}}$ holds for any k ≥ 2, and, if the Riemann's hypothesis is false this inequality holds for infinitely many k, and is false for infinitely many k.     

On the exact asymptotic behaviour of the distribution of ladder epochs

Let T₊ denote the first increasing ladder epoch in a random walk with a typical step-length X. It is known that for a large class of random walks with E(X)=0,E(X²)=∞, and the right-hand tail of the distribution function of X asymptotically larger than the left-hand tail, $\text{P}\text{T}{}_{\mathrm{+}}\text{?n∽n}{}^{\mathrm{<mtext>1</mtext><mtext>\beta ?1</mtext>}}\text{L}{}_{\mathrm{+}}\text{(n)}$ as n→∞, with 1<β<2 and L₊ slowly varying, if and only if$\text{P}\text{{X?x}∽ 1/{x}{}^{\mathrm{\beta }}\text{L(x)}}$ as x→+∞, with L slowly varying. In this paper it is shown how the asymptotic behaviour of L determines the asymptotic behaviour of L₊ and vice versa. As a by-product, it follows that a certain class of random walks which are in the domain of attraction of one-sided stable laws is such that the down-going ladder height distribution has finite mean.     

Largest and smallest maximal sets of pairwise disjoint partitions

Let $\text{X}$ be a maximal set of pairwise disjoint partitions of n into t distinct parts. Let M_t(n) (resp. m_t(n)) denote the size of the largest (resp. smallest) such maximal set $\text{X}$. Upper and lower bounds for $\text{M}{}_{\mathrm{t}}\text{(n)}\text{n}$ and $\text{m}{}_{\mathrm{t}}\text{(n)}\text{n}$ are established.     

Minimum variance estimators for misclassification probabilities in discriminant analysis

Let α(n₁, n₂) be the probability of classifying an observation from population Π₁ into population Π₂ using Fisher's linear discriminant function based on samples of size n₁ and n₂. A standard estimator of α, denoted by T₁, is the proportion of observations in the first sample misclassified by the discriminant function. A modification of T₁, denoted by T₂, is obtained by eliminating the observation being classified from the calculation of the discriminant function. The UMVU estimators, $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$, of ET₁ = τ₁(n₁, n₂) and ET₂ = τ₂(n₁, n₂) = α(n₁ ? 1, n₂) are derived for the case when the populations have multivariate normal distributions with common dispersion matrix. It is shown that $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$ are nonincreasing functions of D², the Mahalanobis sample distance. This result is used to derive the sampling distributions and moments of $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$. It is also shown that α is a decreasing function of Δ² = (μ₁ ? μ₂)′Σ^?1(μ₁ ? μ₂). Hence, by truncating $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$ (or any estimator) at the value of α for Σ = 0, new estimators are obtained which, for all samples, are as close or closer to α.     

Central limit theorems for C(S)-valued random variables

Let C(S) be the space of real-valued continuous functions on a compact metric space S. Let {X_n, n ? 1} be a sequence of independent identically distributed C(S)-valued random variables with mean zero and sup_t?sE[X₁²(t)] = 1. We show that the measures induced by ($\text{X}{}_1\text{+ ··· + X}{}_{\mathrm{n}}\text{) n}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ converge weakly to a Gaussian measure on C(S) under different conditions on X₁, one of which consolidates and extends results of Strassen and Dudley, Giné, and Dudley. Our method of proof is different from the methods employed by these authors.     

Zeta matrices of elliptic curves

Let $\text{O =}\text{lim}{}_{\mathrm{n}}\text{Z/p}{}^{\mathrm{n}}\text{Z}$, let $\text{A}\text{= O[g}{}_2\text{, g}{}_3\text{]}{}_{\mathrm{\Delta }}$, where g₂ and g₃ are coefficients of the elliptic curve: Y² = 4X³ ? g₂X ? g₃ over a finite field and Δ = g₂³ ? 27g₃² and let $\text{B}\text{=}\text{A}\text{[X, Y]}\text{(Y}{}^2\text{? 4X}{}^3\text{+ g}{}_2\text{X + g}{}_3\text{)}$. Then the p-adic cohomology theory will be applied to compute explicitly the zeta matrices of the elliptic curves, induced by the pth power map on the free $\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q}$-module $\text{H}{}^1\text{(X,}\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$. Main results are; Theorem 1.1: X²dY and YdX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{, Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$; Theorem 1.2: YdX, X²dY, Y^?1dX, Y^?2dX and XY^?2dX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{? (Y = 0), Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$, where $\text{X}$ is a lifting of X, and all the necessary recursive formulas for this explicit computation are given.     

Linear operators preserving the (p,q)-numerical range

Let $\text{C}{}_{\mathrm{n\times n}}$ and $\text{H}{}_{\mathrm{n}}$ denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p?q?n. For $\text{A?}\text{C}{}_{\mathrm{n\times n}}$, the (p,q)-numerical range of A is the set

$\text{W}{}_{\mathrm{p,q}}\text{(A)={trC}{}_{\mathrm{p}}\text{(J}{}_{\mathrm{q}}\text{UAU}{}^1\text{):U}\text{unitary}\text{}}$

, where C_p(X) is the pth compound matrix of X, and Jq is the matrix I_q?O_n-q. Let $\text{L}$ denote $\text{H}$_n or $\text{C}{}_{\mathrm{n\times n}}$. The problem of determining all linear operators T: $\text{L}$→$\text{L}$ such that

$\text{W}{}_{\mathrm{p,q}}\text{(T(A))=W}{}_{\mathrm{p,q}}\text{(A)}\text{for all}\text{A?}\text{L}$

is treated in this paper.     

On convergence rates and the expectation of sums of random variables

Let X_i be iidrv's and S_n=X₁+X₂+…+X_n. When EX²₁<+∞, by the law of the iterated logarithm $\text{(S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{)}\text{(n log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{→0 a.s.}$ for some constants α_n. Thus the r.v. $\text{Y=sup}{}_{\mathrm{n?1}}\text{[|S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{|?(δn log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{]}{}^{\mathrm{+}}$ is a.s.finite when δ>0. We prove a rate of convergence theorem related to the classical results of Baum and Katz, and apply it to show, without the prior assumption EX²₁<+∞ that EY^h<+∞ if and only if $\text{E}\text{|X}{}_1\text{|}{}^{\mathrm{2+h}}\text{[log|X}{}_1\text{|]}{}^{-1}\text{<+∞}$ for 0<h<1 and δ> $\text{h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$, whereas $\text{E}\text{Y}{}^{\mathrm{h}}\text{=+∞}$ whenever h>0 and $\text{0<δ<h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$.     

Arnold's canonical matrices and the asymptotic simplification of ordinary differential equations

Let A(x,ε) be an n×n matrix function holomorphic for |x|?x₀, 0<ε?ε₀, and possessing, uniformly in x, an asymptotic expansion $\text{A(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{A}{}_{\mathrm{r}}\text{(x) ε}{}^{\mathrm{r}}$, as ε→0⁺. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion $\text{P(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{P}{}_{\mathrm{r}}\text{(x)ε}{}^{\mathrm{r}}$, as ε→0⁺, is constructed, such that the transformation y = P(x,ε)z takes the differential equation $\text{ε}{}^{\mathrm{h}}\text{dy}\text{dx}\text{= A(x,ε)y,h}$ a positive integer, into $\text{ε}{}^{\mathrm{h}}\text{dz}\text{dx}\text{= B(x,ε)z}$, where B(x,ε) is asymptotically equal, to all orders, to a matrix in a canonical form for holomorphic matrices due to V.I. Arnold.     

Invariant subspaces of certain subnormal operators

Let T be a subnormal, nonnormal operator on a Hilbert space and suppose that the point spectrum of $\text{T}{}^1$ is empty. Then there exist vectors x ≠ 0 for which $\text{(T}{}^1\text{? zI)}{}^{\mathrm{?1}}\text{x}$ exists and is weakly continuous for all z. It is shown that under certain conditions, the Cauchy integral of this vector function taken around an appropriate contour, not necessarily lying in the resolvent set of $\text{T}{}^1$, leads to a proper (nontrivial) invariant subspace of $\text{T}{}^1$.