An existence theorem for antichains

Let k₁, k₂,…, k_n be given integers, 1 ? k₁ ? k₂ ? … ? k_n, and let S be the set of vectors x = (x₁,…, x_n) with integral coefficients satisfying 0 ? x_i ? k_i, i = 1, 2, 3,…, n. A subset H of S is an antichain (or Sperner family or clutter) if and only if for each pair of distinct vectors x and y in H the inequalities x_i ? y_i, i = 1, 2,…, n, do not all hold. Let |H| denote the number of vectors in H, let K = k₁ + k₂ + … + k_n and for 0 ? l ? K let (l)H denote the subset of H consisting of vectors h = (h₁, h₂,…, h_n) which satisfy h₁ + h₂ + … + h_n = l. In this paper we show that if H is an antichain in S, then there exists an antichain H′ in S for which |(l)H′| = 0 if $\text{l <}\text{K}\text{2}$, $\text{|(}\text{K}\text{2}\text{)H′| = |(}\text{K}\text{2}\text{)H|}$ if K is even and |(l)H′| = |(l)H| + |(K ? l)H| if $\text{l>}\text{K}\text{2}$.     

Prime interchange graphs of classes of matrices of zeros and ones

Let R = (r₁,…, r_m) and S = (s₁,…, s_n) be nonnegative integral vectors, and let $\text{U}$(R, S) denote the class of all m × n matrices of 0's and 1's having row sum vector R and column sum vector S. An invariant position of $\text{U}$(R, S) is a position whose entry is the same for all matrices in $\text{U}$(R, S). The interchange graph G(R, S) is the graph where the vertices are the matrices in $\text{U}$(R, S) and where two matrices are joined by an edge provided they differ by an interchange. We prove that when 1 ≤ r_i ≤ n ? 1 (i = 1,…, m) and 1 ≤ s_j ≤ m ? 1 (j = 1,…, n), G(R, S) is prime if and only if $\text{U}$(R, S) has no invariant positions.     

On a problem of Katona on minimal completely separating systems with restrictions

Let S be a set of n elements, and k a fixed positive integer $\text{<}\text{1}\text{2}\text{n}$. Katona's problem is to determine the smallest integer m for which there exists a family $\text{A}$ = {A₁, …, A_m} of subsets of S with the following property: |_i| ? k (i = 1, …, m), and for any ordered pair x_i, x_i ∈ S (i ≠ j) there is A₁ ∈ $\text{A}$ such that x_i ∈ A₁, x_j ? A₁. It is given in this note that $\text{m = ?}\text{2n}\text{k}\text{?}\text{if}\text{1}\text{2}\text{k}{}^2\text{? 2}$.     

Some first passage problems related to cusum procedures

Let $\text{X}{}_1\text{,}\text{X}{}_2\text{,…}$ be independent random variables, and set W_n = max(0,W_n-1 + X_n), W₀ = 0, n ? 1. The so-called cusum (cumulative sum) procedure uses the first passage time T(h) = inf{n ? 1: W_n?h}for detecting changes in the mean μ of the process. It is shown that lim_h→∞ μET(h)/h = 1 if μ > 0. Also, a cusum procedure for detecting changes in the normal mean is derived when the variance is unknown. An asymptotic approximation to the average run length is given.     

Maximization,under equality constraints,of a functional of a probability distribution

Let $\text{F}$ be a family of probability distributions. Let O, C₁…C_n be real functions on $\text{F}$. Let z₁…z_n be real numbers. Then we consider the problem of maximization of the object function O(F)(F?$\text{F}$) under the equality constraints C₁(F)=z₁(i=1,…,n) . The theory is developed in order to solve problems of the following kind: Find the maximal variance of a stop-loss reinsured risk under partial information on the risk such as its range and two first moments.     

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 → ∞.     

On moment conditions for normed sums of independent variables and martingale differences

Let X₁, X₂,… be a sequence of i.i.d. random variables and S_n their partial sums. Necessary and sufficient conditions are given for $\text{{n}{}^{\mathrm{<mtext>?1</mtext><mtext>q</mtext>}}\text{S}{}_{\mathrm{n}}\text{}}{}^{\mathrm{\propto }}{}_1$ to have uniformly bounded pth moments, 0<p<q?2.Some of the results are generalized to martingle differences.     

Definition and characterization of multivariate negative binomial distribution

The probability generating function (pgf) of an n-variate negative binomial distribution is defined to be [β(s₁,…,s_n)]^?k where β is a polynomial of degree n being linear in each s_i and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h(x₁,…,x_n)$\text{exp}\text{(}\text{Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{θ}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}\text{)}\text{f(θ}{}_1\text{,…,θ}{}_{\mathrm{n}}\text{)}$ is negative binomial if and only if its univariate marginals are negative binomial. Let S_t, t = 1,…, m, be subsets of {s₁,…, s_n} with empty ∩_t=1^mS_t. Then an n-variate pgf is of a negative binomial if and only if for all s in S_t being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t.     

Linear transformations on matrices: the invariance of generalized permutation matrices—III

Let F be a field, F1 be its multiplicative group, and $\text{H}$ = {H:H is a subgroup of F1 and there do not exist a, b?F1 such that Ha+b?H}. Let D_n be the dihedral group of degree n, H be a nontrivial group in $\text{H}$, and τ_n(H) = {α = (α₁, α₂,…, α_n):α_i?H}. For σ?D_n and α?τ_n(H), let P(σ, α) be the matrix whose (i,j) entry is α_iδ_iσ(j) (i.e., a generalized permutation matrix), and

$\text{P(D}{}_{\mathrm{n}}\text{, H) = {P(σ, α):σ?D}{}_{\mathrm{n}}\text{, α?τ}{}_{\mathrm{n}}\text{(H)}}$

. Let M_n(F) be the vector space of all n×n matrices over F and $\text{T}$P(D_n, H) = {T:T is a linear transformation on M_n (F) to itself and T(P(D_n, H)) = P(D_n, H)}. In this paper we classify all T in $\text{T}$P(D_n, H) and determine the structure of the group $\text{T}$P(D_n, H) (Theorems 1 to 4). An expository version of the main results is given in Sec. 1, and an example is given at the end of the paper.     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

Diagonal norm hermitian matrices

If v is a norm on $\text{C}$ⁿ, let H(v) denote the set of all norm-Hermitians in $\text{C}$ⁿⁿ. Let S be a subset of the set of real diagonal matrices D. Then there exists a norm v such that S=H(v) (or S = H(v)∩D) if and only if S contains the identity and S is a subspace of D with a basis consisting of rational vectors. As a corollary, it is shown that, for a diagonable matrix h with distinct eigenvalues λ₁,…, λ_r, r?n, there is a norm v such that h ∈ H(v), but h^s?H(v), for some integer s, if and only if λ₂–λ₁,…, λ_r–λ₁ are linearly dependent over the rationals. It is also shown that the set of all norms v, for which H(v) consists of all real multiples of the identity, is an open, dense subset, in a natural metric, of the set of all norms.     

On the irreducibility of multivariate subresultants

Let P₁,…,P_n be generic homogeneous polynomials in n variables of degrees d₁,…,d_n respectively. We prove that if ν is an integer satisfying ∑_i=1ⁿd_i?n+1?min{d_i}<ν, then all multivariate subresultants associated to the family P₁,…,P_n in degree ν are irreducible. We show that the lower bound is sharp. As a byproduct, we get a formula for computing the residual resultant of $\text{ρ?ν+n?1}\text{n?1}$ smooth isolated points in $\text{P}{}^{\mathrm{<mspace\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>n?1</mtext>}}\text{.}$To cite this article: L. Busé, C. D'Andrea, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The asymptotic ruin problem when the healthy and sick periods form an alternating renewal process

Let {T₁, Y₁}^∞_i=1 be a sequence of positive independent random variables. Let, also, Z₁ = βY′₁ ? πT_i, i = 1, 2, …, where Y′₁ = Max(0, Y_i ? w), w ? 0, and where β < 0 and π is such that E(Z₁) < 0. We consider the random walk of partial sums S_n = ?ⁿ_i=1Z_i in the presence of an absorbing region (u, ∞), u ? 0, and S₀ ≡ 0. Of interest is $\text{ψ(u) = Pr(}\text{S}\text{?}\text{≤ u)}$ where S? = Sup(0, S₁, S₂, …, S_n, …).     

On maximal antichains containing no set and its complement

Let 1?k₁?k₂?…?k_n be integers and let S denote the set of all vectors x = (x₁, …, x_n with integral coordinates satisfying 0?x_i?k_i, i = 1,2, …, n; equivalently, S is the set of all subsets of a multiset consisting of k_i elements of type i, i = 1,2, …, n. A subset X of S is an antichain if and only if for any two vectors x and y in X the inequalities x_i?y_i, i = 1,2, …, n, do not all hold. For an arbitrary subset H of S, (i)H denotes the subset of H consisting of vectors with component sum i, i = 0, 1, 2, …, K, where K = k₁ + k₂ + …k_n. |H| denotes the number of vectors in H, and the complement of a vector x?S is (k₁-x₁, k₂-x₂, …, k_n -x_n). What is the maximal cardinality of an antichain containing no vector and its complement? The answer is obtained as a corollary of the following theorem: if X is an antichain, K is even and$\text{|(}\text{1}\text{2}\text{K)X|}$ does not exceed the number of vectors in $\text{(}\text{1}\text{2}\text{K)S}$ with first coordinate different from k₁, then

$\text{∑}\text{i=0}\text{K}\text{i≠}\text{1}\text{2}\text{K}\text{|(i)X|}\text{|(i)S|}\text{+}\text{|(}\text{1}\text{2}\text{K)X|}\text{|(}\text{1}\text{2}\text{K-1)S|}\text{?1}$

.     

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).     

Intersection theorems for group divisible difference sets

Let D be an (m,n;k;λ₁,λ₂)-group divisible difference set (GDDS) of a group G, written additively, relative to H, i.e. D is a k-element subset of G, H is a normal subgroup of G of index m and order n and for every nonzero element g of G,?{(d₁,d₂)?,d₁,d₂?D,d₁?d₂=g}? is equal to λ₁ if g is in H, and equal to λ₂ if g is not in H. Let H₁,H₂,…,H_m be distinct cosets of H in G and S_i=D∩H_i for all i=1,2,…,m. Some properties of S₁,S₂,…,S_m are studied here. Table 1 shows all possible cardinalities of S_i's when the order of G is not greater than 50 and not a prime. A matrix characterization of cyclic GDDS's with λ₁=0 implies that there exists a cyclic affine plane of even order, say n, only if n is divisible by 4 and there exists a cyclic (n?1,$\text{1}\text{2}$n?1,$\text{1}\text{4}$n?1)-difference set.     

Local asymptotic normality for progressively censored likelihood ratio statistics and applications

Let X_n,1 ≤ X_n,2 ≤ … ≤ X_n,n be the ordered variables corresponding to a random sample of size n with respect to a family of probability measures {P_θ:θ ∈ Θ} where Θ is an open subset of the real line. In many practical situations the X_n,i are the observables and experimentation must be curtailed prior to X_n,n. If τ_n is a stopping variable adapted to the σ-fields {σ(X_n,1,…,X_n,k): 1 ≤ k ≤ n} and P_n,θ the projection of P_θ onto σ(X_n,1,…,X_n,τn), the local asymptotic normality of the stopped progressively censored likelihood ratio statistics is established with θ, $\text{θ}{}_{\mathrm{n}}\text{= θ + un}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{∈ Θ}$ and θ, u held fixed, under certain conditions on the underlying distribution and on τ_n. Conditions are also given to ensure the local asymptotic normality of likelihood ratio statistics where the underlying observations are given in a series scheme.     

Note on lower bounds for the rank of a matrix

The following results are proved: Let A = (a_ij) be an n × n complex matrix, n ? 2, and let k be a fixed integer, 1 ? k ? n ? 1.(1) If there exists a monotonic G-function f = (f₁,…,f_n) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have

$\text{Π}\text{i∈S}\text{f}{}_{\mathrm{i}}\text{(A)<}\text{Π}\text{i∈S}\text{|a}{}_{\mathrm{ii}}\text{|,}$

then the rank of A is ? n ? k + 1. (2) If A is irreducible and if there exists a G-function f = (f₁,…,f_n) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have

$\text{Π}\text{i∈S}\text{f}{}_{\mathrm{i}}\text{(A)<}\text{Π}\text{i∈S}\text{|a}{}_{\mathrm{ii}}\text{|,}$

then the rank of A is ? n ? k + 1 if k ? 2, n ? 3; it is ? n ? 1 if k = 1.     

A note on the J1-convergence of a first-passage process

Let X₁, X₂, … be a sequence of independent and identically distributed random variables with mean zero such that the common distribution function belongs to the domain of attraction of a stable law G_α,β with 1<α<2 and β=1 or α=2. If S_n=X₁+…X_n and N(ξ)=min{k:S_k>ξ}, ξ>0, then it is shown that $\text{N(nt)}\text{B}{}^1\text{(n)}$, 0<t<1, converges weakly under the Skorohod J₁-topology to a stable subordinator of index $\text{1}\text{α}$, where B¹(n) depends on the norming constant for S_n.     

The covering problem for the fields Q(−1)12 and Q(−3)12

Let k be a positive square free integer, $\text{N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ the ring of algebraic integers in $\text{Q(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and S the unit sphere in C_n, complex n-space. If A₁,…, A_n are n linearly independent points of C_n then L = {u₁Au + … + u_nA_n} with $\text{u}{}_{\mathrm{r}}\text{∈ N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is called a k-lattice. The determinant of L is denoted by d(L). If L is a covering lattice for S, then $\text{θ(S, L) =}\text{V(S)}\text{d(L)}$ is the covering density. L is called locally (absolutely) extreme if θ(S, L) is a local (absolute) minimum. In this paper we determine unique classes of extreme lattices for k = 1 and k = 3.