The expected eigenvalue distribution of a large regular graph

Let K₁, K₂,... be a sequence of regular graphs with degree v?2 such that n(X_i)→∞ and ck(X_i)/n(X_i)→0 as i∞ for each k?3, where n(X_i) is the order of X_i, and c_k(X_i) is the number of k- cycles in X₁. We determine the limiting probability density f(x) for the eigenvalues of X>_i as i→∞. It turns out that

$\text{f(x)=}\text{v}\text{4(v?1)?v}{}^2\text{2π(v}{}^2\text{?x}{}^2\text{)}{}_0$

for ?x??2$\text{v-1}$, otherwise It is further shown that f(x) is the expected eigenvalue distribution for every large randomly chosen labeled regular graph with degree v.     

Identifiability of the multinormal and other distributions under competing risks model

Let X₁, X₂ ,…, X_p be p random variables with joint distribution function F(x₁ ,…, x_p). Let Z = min(X₁, X₂ ,…, X_p) and I = i if Z = X_i. In this paper the problem of identifying the distribution function F(x₁ ,…, x_p), given the distribution Z or that of the identified minimum (Z, I), has been considered when F is a multivariate normal distribution. For the case p = 2, the problem is completely solved. If p = 3 and the distribution of (Z, I) is given, we get a partial solution allowing us to identify the independent case. These results seem to be highly nontrivial and depend upon Liouville's result that the (univariate) normal distribution function is a nonelementary function. Some other examples are given including the bivariate exponential distribution of Marshall and Olkin, Gumbel, and the absolutely continuous bivariate exponential extension of Block and Basu.     

Upper bounds on stop-loss premiums in case of known moments up to the fourth order

In actuarial sciences recently a lot of results have been derived for solving the problem sup {E(X−t) +:r.υ. X >0,EX′ = μ, for i = 1, 2, …, k}, x where μ, i = 1 to k as well as t are given. The present contribution solves this problem up to k = 4 analytically.     

Some path properties of generalized Lévy sheet

Let X(t) be an N parameter generalized Lévy sheet taking values in ℝ^d with a lower index α, ℜ = {(s, t] = ∏ _i=1 ^N (s _i, t _i], s _i < t _i}, E(x, Q) = {t ∈ Q: X(t) = x}, Q ∈ ℜ be the level set of X at x and X(Q) = {x: ∃t ∈ Q such that X(t) = x} be the image of X on Q. In this paper, the problems of the existence and increment size of the local times for X(t) are studied. In addition, the Hausdorff dimension of E(x, Q) and the upper bound of a uniform dimension for X(Q) are also established.     

Reconstruction from vertex-switching

Let X be a graph with vertices x₁ ,…, x_n. Let X_i be the graph obtained by removing all edges {x_i, x_j} of X and inserting all nonedges {x_i, x_k}. If n ? 0 (mod 4), then X can be uniquely reconstructed from the unlabeled graphs X₁.…, X_n. If n = 4 the result is false, while for n = 4m≥8 the result remains open. The proof uses linear algebra and does not explicitly describe the reconstructed graph X.     

A generalized coupon collecting model as a parsimonious optimal stochastic assignment model

There is a given set of n boxes, numbered 1 thru n. Coupons are collected one at a time. Each coupon has a binary vector x ₁,…,x _n attached to it, with the interpretation being that the coupon is eligible to be put in box i if x _i =1,i=1…,n. After a coupon is collected, it is put in a box for which it is eligible. Assuming the successive coupon vectors are independent and identically distributed from a specified joint distribution, the initial problem of interest is to decide where to put successive coupons so as to stochastically minimize N, the number of coupons needed until all boxes have at least one coupon. When the coupon vector X ₁,…,X _n is a vector of independent random variables, we show, if P(X _i =1) is nondecreasing in i, that the policy π that always puts an arriving coupon in the smallest numbered empty box for which it is eligible is optimal. Efficient simulation procedures for estimating P _π (N>r) and E _π [N] are presented; and analytic bounds are determined in the independent case. We also consider the problem where rearrangements are allowed.     

On the structure of cluster point sets of iteratively generated sequences

LetX be a closed subset of a topological spaceF; leta(·) be a continuous map fromX intoX; let {x _i} be a sequence generated iteratively bya(·) fromx ₀ inX, i.e.,x _i+1 =a(x _i),i=0, 1, 2, ...; and letQ(x ₀) be the cluster point set of {x _i}. In this paper, we prove that, if there exists a pointz inQ(x ₀) such that (i)z is isolated with respect toQ(x ₀), (ii)z is a periodic point ofa(·) of periodp, and (iii)z possesses a sequentially compact neighborhood, then (iv)Q(x ₀) containsp points, (v) the sequence {x _i} is contained in a sequentially compact set, and (vi) every point inQ(x ₀) possesses properties (i) and (ii). The application of the preceding results to the caseF=E ⁿ leads to the following: (vii) ifQ(x ₀) contains one and only one point, then {x _i} converges; (viii) ifQ(x ₀) contains a finite number of points, then {x _i} is bounded; and (ix) ifQ(x ₀) containsp points, then every point inQ(x ₀) is a periodic point ofa(·) of periodp.     

Sufficient conditions for existence of extrema in linear programming

It is shown that, on a closed convex subset X of a real Hausdorff locally convex space E, a continuous linear functional x′ on E has an extremum at an extreme point of X, provided X contains no line and X ∩ (x′)^?1 (λ₀) is non-empty and weakly compact for some real λ₀. It is also shown that any weakly locally compact closed convex subset of E that contains no line is the sum of its asymptotic cone and the closed convex hull of its extreme points.     

Waldhausen's theory of k-fold end structures: a survey

Let R⁺ be the space of nonnegative real numbers. F. Waldhausen defines a k-fold end structure on a space X as an ordered k-tuple of continuous maps x_f:X → R⁺, 1 ? j ? k, yielding a proper map x:X → (R⁺)^k. The pairs (X,x) are made into the category E^k of spaces with k-fold end structure. Attachments and expansions in E^k are defined by induction on k, where elementary attachments and expansions in E⁰ have their usual meaning. The category E^k/Z consists of objects (X, i) where i: Z → X is an inclusion in E^k with an attachment of i(Z) to X, and the category E^k6Z consists of pairs (X,i) of E^k/Z that admit retractions X → Z. An infinite complex over Z is a sequence X = {X₁ ? X₂ ? … ? X_n …} of inclusions in E^k6Z. The abelian grou p S₀(Z) is then defined as the set of equivalence classes of infinite complexes dominated by finite ones, where the equivalence relation is generated by homotopy equivalence and finite attachment; and the abelian group S₁(Z) is defined as the set of equivalence classes of X₁, where X ∈ E^k/Z deformation retracts to Z. The group operations are gluing over Z. This paper presents the Waldhausen theory with some additions and in particular the proof of Waldhausen's proposition that there exists a natural exact sequence 0 → S₁(Z × R)→^πS₀(Z) by utilizing methods of L.C. Siebenmann. Waldhausen developed this theory while seeking to prove the topological invariance of Whitehead torsion; however, the end structures also have application in studying the splitting of a noncompact manifold as a product with R[1].     

An improved approximation algorithm for requirement cut

This note presents improved approximation guarantees for the requirement cut problem: given an n-vertex edge-weighted graph G=(V,E), and g groups of vertices X₁,…,X_g⊆V with each group X_i having a requirement r_i between 0 and |X_i|, the goal is to find a minimum cost set of edges whose removal separates each group X_i into at least r_i disconnected components. We give a tight Θ(logg) approximation ratio for this problem when the underlying graph is a tree, and show how this implies an O(logk⋅logg) approximation ratio for general graphs, where .     

Sphere of influence graphs in general metric spaces

Let χ = {X₁…, X_n} be a set of points in a metric space (n ≥ 2). Let r_i, denote the minimum distance between X_i and the other points in χ. The sphere of influence at X_i is the open ball with center X_iand radius r_i. The sphere of influence graph has vertex set χ with an edge joining a pair of distinct vertices provided the corresponding spheres of influence intersect. This proximity graph was introduced by Toussaint to model computer vision and pattern recognition problems in the Euclidean plane. We discuss the abstract properties of sphere of influence graphs in general metric spaces with emphasis on normed linear spaces. Our work generalizes and simplifies some theorems in the literature on sphere of influence graphs and lays the groundwork for future work.     

Stability of Friedrichs's scheme in the maximum norm for hyperbolic systems in one space dimension

We are concerned with Friedrichs's scheme for an initial value problem u_t(t, x) = A(t, x)u_x(t, x), u(0, x) = u₀(x), where u₀(x) belongs to L_∞, not to L₂. We show that Friedrichs's scheme is stable in the maximum norm ·_L∞, provided that the system is regularly hyperbolic and that the eigenvalues d_i(t, x) (i = 1,2,..., N) of the N XN matrix A(t, x) satisfy the conditions 1±λd_i(t, x)?0 (i = 1,2,..., N), where λ is a mesh ratio.     

Optimum designs when the observations are second-order processes

Let the process {Y(x,t) : t?T} be observable for each x in some compact set X. Assume that Y(x, t) = θ₀f₀(x)(t) + … + θ_kf_k(x)(t) + N(t) where f_i are continuous functions from X into the reproducing kernel Hilbert space H of the mean zero random process N. The optimum designs are characterized by an Elfving's theorem with $\text{R}$ the closed convex hull of the set {(φ, f(x))_H : 6φ 6_H ≤ 1, x?X}, where (·, ·)_H is the inner product on H. It is shown that if X is convex and f_i are linear the design points may be chosen from the extreme points of X. In some problems each linear functional c′θ can be optimally estimated by a design on one point x(c). These problems are completely characterized. An example is worked and some partial results on minimax designs are obtained.     

Caterpillar arboricity of planar graphs

We solve a conjecture of Roditty, Shoham and Yuster [P.J. Cameron (Ed.), Problems from the 17th British Combinatorial Conference, Discrete Math., 231 (2001) 469-478; Y. Roditty, B. Shoham, R. Yuster, Monotone paths in edge-ordered sparse graphs, Discrete Math. 226 (2001) 411-417] on the caterpillar arboricity of planar graphs. We prove that for every planar graph G=(V,E), the edge set E can be partitioned into four subsets (E_i)_1?i?4 in such a way that G[E_i], for 1?i?4, is a forest of caterpillars. We also provide a linear-time algorithm which constructs for a given planar graph G, four forests of caterpillars covering the edges of G.     

Familles de graphes expanseurs et paires de HeckeFamilies of expanding graphs and Hecke pairs

Let H be a subgroup of a group G. Suppose that (G,H) is a Hecke pair and that H is finitely generated by a finite symmetric set of size k. Then G/H can be seen as a graph (possibly with loops and multiple edges) whose connected components form a family (X_i)_i∈I of finite k-regular graphs. In this Note, we analyse when the size of these graphs is bounded or tends to infinity and we present criteria for (X_i)_i∈I to be a family of expanding graphs as well as some examples. To cite this article: M.B. Bekka et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 463–468.     

Probabilities of maximal deviations for nonparametric regression function estimates

Let (X, Y) have regression function m(x) = E(Y | X = x), and let X have a marginal density f₁(x). We consider two nonparameteric estimates of m(x): the Watson estimate when f₁ is known and the Yang estimate when f₁ is known or unknown. For both estimates the asymptotic distribution of the maximal deviation from m(x) is proved, thus extending results of Bickel and Rosenblatt for the estimation of density functions.     

Classes of matrices associated with the optimal assignment problem

Let E=[e_ij] be a matrix with integral elements, and let x be an indeterminate defined over the rational field Q. We investigate matrices of the form X=[x^e_ij] (i = 1,…, m; j = 1,…, n; m ⩽ n). We may multiply the lines (rows or columns) of the matrix X by suitable integral powers of x in various ways and thereby transform X into a matrix Y=[x^f_ij] such that the f_ij are nonnegative integers and each line of Y contains at least one element x⁰ = 1. We call Y a normalized form of X, and we denote by S(X) the class of all normalized forms associated with a given matrix X. The classes S(X) have a fascinating combinatorial structure, and the present paper is a natural outgrowth and extension of an earlier study. We introduce new concepts such as an elementary transformation called an interchange. We prove, for example, that two matrices in the same class are transformable into one another by interchanges. Our analysis of the class S(X) also yields new insights into the structure of the optimal assignments of the matrix E by way of the diagonal products of the matrix X.     

On the cumulative quantile regression process

Let (X, Y) be a bivariate random vector and F(x) the marginal distribution function of X. The quantile regression (QR) function of Y on X is defined as r(u) = E[Y | F(X) = u] and the cumulative QR function (CQR) M(u) as its integral over [0, u]. The empirical counterpart based on a sample of size n is M _n (u). In this paper, we construct strong Gaussian approximations of the associated CQR process under appropriate assumptions. The construction provides a firm basis for the study of functional statistics based on M _in (u). A law of the iterated logarithm for the CQR process follows from our result.     

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).     

Stop rule and supremum expectations of i.i.d. random variables: A complete comparison by conjugate duality

A complete comparison is made between the value V(X₁,…, X_n) = sup{EX_t: t is a stop rule for X₁,…,X_n} and E(max_{j ≤ n}X_j) for all uniformly bounded sequences of i.i.d. random variables X₁, …, X_n. Specifically, the set of ordered pairs {(x,y): x = V(X₁, …, X_n) and y = E(max_j≤nX_j) for some i.i.d.r.v.'s X₁,…, X_n taking values in [0, 1]} is precisely the set {(x, y): x≤y≤Γ_n(x); 0 ≤x≤1}, where the upper boundary function Γ_n is given in terms of recursively defined functions. The result yields families of inequalities for the “prophet” problem, relating the motal's value of a game V(X₁, …, X_n) to the prophet's value of the game E(max_j≤nX_j). The proofs utilize conjugate duality theory, probabilistic convexity arguments, and functional equation analysis. Asymptotic analysis of the “prophet” regions and inequalities is also given.