The distribution of the characteristic roots ofS 1 S 2 −1 under violations in the complex case and power comparisons of four tests

Summary The joint density function of the latent roots ofS ₁ S ₂ ⁻¹ under violations is obtained whereS ₁ has a complex non-central Wishart distributionW _c (p,n ₁,Σ ₁,Ω) andS ₂, an independent complex central Wishart,W _c (p,n ₂,Σ ₂, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical.     

Inequalities and sphere-packing inl p

If 1<p<∞, there is a constantr _p <1/2 so that ifr>r _p only a bounded number of balls inl _p of radiusr can be packed into the unit ball ofl _p . We obtain the exact value of this bound for eachp andr as a consequence of several new inequalities relating the expressions Σλ _i λ _j ‖x _i –x _j ‖ ^p , Σλ _i ‖x _i ‖ ^p and Σλ _i ^/2 for sequences (x _i ) ₁ ⁿ ⊂l _p and (λ _i ) ₁ ⁿ ⊂R.     

Connectedness of certain random graphs

L. Dubins conjectured in 1984 that the graph on vertices {1, 2, 3, ...} where an edge is drawn between verticesi andj with probabilityp _ij=λ/max(i, j) independently for each pairi andj is a.s. connected forλ=1. S. Kalikow and B. Weiss proved that the graph is a.s. connected for anyλ>1. We prove Dubin’s conjecture and show that the graph is a.s. connected for anyλ>1/4. We give a proof based on a recent combinatorial result that forλ≦1/4 the graph is a.s. disconnected. This was already proved forλ<1/4 by Kalikow and Weiss. Thusλ=1/4 is the critical value for connectedness, which is surprising since it was believed that the critical value is atλ=1.     

Extremal eigenvalue problems for convex sets of symmetric matrices and operators

LetA ₁,...,A_n andK bem×m symmetric matrices withK positive definite. Denote byC the convex hull of {A ₁,...A_n}. Let {λ _p (KA)} ₁ ⁿ be then real eigenvalues ofKA arranged in decreasing order. We show that maxλ _p (KA) onC is attained for someA ^*=Σ _i ^{= 1/n} for which at mostp(p+1)/2 of α _i ^* do not vanish. We extend this result in several directions and consider applications to classes of integral equations. This paper is based mainly on the author’s doctoral dissertation written at the Technion—Israel Institute of Technology, March 1971, under the direction of Professor B. Schwarz. I wish to thank Professor Schwarz for his advice and encouragement. I am also grateful to Professor S. Karlin for supplying simplifications of several of my arguments. Some extensions discussed here are joint results of Karlin and the author.     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

Regularity properties of solutions of some abstract parabolic equations

Consider the equation (i) (da/dt)—A(t)u(t)=f(t) where fort ∈ [a, b],A(t) is a densely defined and closed linear operator in a Banach spaceX. Assume the existence of bounded projectionsE _i(t),i=1, 2, such thatA(t) E ₁(t) and —A(t)E ₂(t) are infinitesimal generators of analytic semigroups andA(t) is completely reduced by the direct sum decompositionX = Σ _i ^{b = 1/2} ⊕E _i (t)X. We show that any solutionu(t) of (i) is inC ^∞(a, b) and satisfies the inequalities (1.2) provided thatf(t) andA(t) are infinitely differentiable in [a, b] in a suitable sense. In caseA(t) andf(t) are in a Gevrey class determined by the constants {M _n} we have (1. 3). Applications are given to the study of solution of (i) where fort ∈ [a, b]A(t) is the unbounded operator inH ^0,p (G) associated with an elliptic boundary value problem that satisfies Agmon’s conditions on the rays λ=±iτ, τ > 0. Research partially supported by an N.S.F. grant at Brandeis University.     

A sieve result for Farey fractions

This paper is concerned with the sieve problem for Farey fractions (i.e., rational numbers with denominators less thanx) lying in an interval (λ₁, λ₂). An asymptotic formula for the sifting function is derived under the assumption that (λ₁, λ₂)x→∞ asx→∞. Two applications of this result are made. In the first one, the value distribution of the vector η(m/n)=(ξ(m), ξ(n)) is considered; here, fork=p ₁ p ₂...p _s ,p ₁≥p ₂>-..., ξk)_is defined by ξ(k)=(logp ₁/logk, logp ₂/logk,..., logp _s /logk, 0, ...); allp _i are prime numbers. It is shown that the limit distribution is π×π, where π is the Poisson-Dirichlet distribution. The asymptotical behavior of finite-dimensional distributions of ξ(k) for natural numbers was studied by Billingsley, Knuth, Trabb Pardo, Vershik, and others; the result of weak convergence to the Poisson-Dirichlet distribution appears in Donnelly and Grimmett. The second application is concerned with the density of sets {m/n: f(m/n)=a}, wheref is a function with the almost squareful kernel. Supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2600, Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 108–127, January–March, 1999. Translated by V. Stakénas     

Generating Uniform Random Vectors in Z p k : The General Case

We consider the rate of convergence of the Markov chain X _n+1=A X _n +B _n (mod p), where A is an integer matrix with nonzero eigenvalues, and {B _n } _n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Q ^k invariant under A. If for all eigenvalues λ _i of A, then n=O((ln p)²) steps are sufficient and n=O(ln p) steps are necessary to have X _n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λ _i that are roots of positive integer numbers, |λ ₁|=1 and |λ _i |>1 for all , then O(p ²) steps are necessary and sufficient.      

Large-deviation probability and the local dimension of sets

Let X ₁ , X ₂ , ..., X_n be n independent identically distributed real random variables and S_n = Σ _n=1 ⁿ X_i. We obtain precise asymptotics forP (S_n ∈ nA) for rather arbitrary Borel sets A₁ in terms of the density of the dominating points in A. Our result extends classical theorems in the field of large deviations for independent samples. We also obtain asymptotics forP (S_n ∈ γ_nA), with γ_n/n → ∞. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

On the exact non-null distribution of Wilks'L VC criterion and power studies

Letx ₁, x₂, ..., x_Nbep×1 random vectors distributed independently asN(u, Σ), Σ>0;u and Σ are unknown. In this paper, we derive the exact non-null distribution of Wilks' likelihood ratio criterion,L _VC, for testingH:∑=σ ²[(1−ρ)I+ρee′], σ>0 and ρ are unknown against the alternativeA≠H,e′=(1, 1, …, 1): 1×p. The distribution has been derived in three series forms: (1) a series of Meijer'sG-functions through Mellin transform, (2) an, alternate series using contour, intergration and (3) a series of chi square distributions. Powers have been computed based on these forms of the distribution forp=2 and 3.     

Speed of convergence in nonparametric kernel estimation of a regression function and its derivatives

Summary The objective in nonparametric regression is to infer a functiong(x) and itspth order derivativesg ^(g)(x),p≧1 fixed, on the basis of a finite collection of pairs {x _i, g(x_i)+Z _i} _i=1 ⁿ , where the noise componentsZ _i satisfy certain modest assumptions and the domain pointsx _i are selected non-randomly. This paper exhibits a new class of kernel estimatesg _n ^(p) ,p≧0 fixed. The main theoretical results of this study are the rates of convergence obtained for mean square and strong consistency ofg _n ^(p) each of them being uniform on the (0,1).     

A result on rearrangements

Circular symmetry is defined for ordered sets ofn real numbers: (y)=(y ₁,...,y _n). Letf(x) be non-decreasing and convex forx≧0 and let (y) be given except in arrangement. The Σ _i ⁼¹ⁿ f(|y _i−y _i+1|) (wherey _n+1=y ₁) is minimal if (and under some additional assumptions only if) (y) is arranged in circular symmetrical order. Sponsored by the Mathematics Research Center, United States Army under Contract No. DA-11-022-ORD-2059, University of Wisconsin, Madison.     

Topological classification of holomorphic, semi-hyperbolic germs, in ``Quasi-Absence' of resonances

We give the classification, under topological conjugacy, of invertible holomorphic germs f:, with λ₁, . . . ,λ_n eigenvalues of d f₀, and |λ_i|≠1 for i=2, . . . ,n while λ₁ is a root of the unity, in the suitable hypothesis of ``quasi-absence' of resonances (i.e., assuming that for r_i≥0 and i=2, . . . ,n, with ).     

Small prime solutions of a pair of linear equations in five prime variables

Assume an additional congruent condition on the coefficients. We prove that the pair 5 of linear equations ∑j=1^5 αλjpj = bλ （λ= 1, 2） has solutions in primes pj satisfying pj 〈〈 （｜b1｜＋｜b2｜＋1） maxλ,j ｜αλj｜^2318＋ε. This improves the exponent 79680 without assuming the additional condition of the second author＇s.     

The total reducibility order of a polynomial in two variables

LetK be an algebraically closed field of characteristic zero. ForA ∈K[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏ _i=1 ^n(λ) A _iλ ^k _μ whereA _iλ are distinct primes. Let ϱ_λ(A) =n(λ) − 1 and let ρ(A) = Σ_λɛσ(A)ϱ_λ(A). The main result is the following: Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA.     

Fractional Poisson equations and ergodic theorems for fractional coboundaries

For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT _α=Σ _j=1 ^∞ a _j T ^j , where {a _j } are the coefficients in the power series expansion (1-t)^α=1-Σ _j=1 ^∞ a _j t ^j in the open unit disk, which satisfya _j >0 anda _j >0 and Σ _j=1 ^∞ a _j =1. The operator calculus justifies the notation(I−T) ^α :=I−T _α (e.g., (I−T _1/2)²=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) ^α x=y, i.e.,y is a coboundary forT _α . The fractional Poisson equation forT is the Poisson equation forT _α . We show that if(I−T)X is not closed, then(I−T) ^α X strictly contains(I−T)X (but has the same closure). ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ _k=1 ^∞ T ^k y/k ^1-α converges in norm, and conclude that lim _n ‖(1/n ^1-α)Σ _k=1 ⁿ T ^k y‖=0 for suchy. For a Dunford-Schwartz operatorT onL ₁ of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T) ^α L ₁ for some 0<α<1, then the one-sided Hilbert transform Σ _k=1 ^∞ T ^k f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T) ^α L _p with α>1−1/p=1/q, then Σ _k=1 ^∞ T ^k f/k ^1/p converges a.e., and thus (1/n ^1/p ) Σ _k=1 ⁿ T ^k f converges a.e. to zero. Whenf∈(I−T) ^1/q L _p (the case α=1/q), we prove that (1/n ^1/p (logn)^1/q )Σ _k=1 ⁿ T ^k f converges a.e. to zero.     

Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums Sn = ∑i=1n Xi,n≥1; S(t) = ∑i=1N(t) Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} are independent non-negative random variables with distribution, Fn, of Xn, independent of {N(t); t≥0}. Special attention is paid to the distribution of dominated variation.     

Bernoulli convolutions and an intermediate value theorem for entropies of<Emphasis Type="Italic">K</Emphasis>-partitions

We establish a strong regularity property for the distributions of the random sums Σ±λ ⁿ , known as “infinite Bernoulli convolutions”: For a.e. λ ∃ (1/2, 1) and any fixed ℓ, the conditional distribution of (w _n+1...,w _n+ℓ) given the sum Σ _n=0 ^∞ w _n λ ⁿ , tends to the uniform distribution on {±1}^ℓ asn → ∞. More precise results, where ℓ grows linearly inn, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system of entropyh hasK-partitions of any prescribed conditional entropy in [0,h]. This answers a question of Rokhlin and Sinai from the 1960’s, for the case of Bernoulli systems. The authors were partially supported by NSF grants DMS-9729992 (E. L.), DMS-9803597 (Y. P.) and DMS-0070538 (W. S.).     

Asymptotic formulas for the hyperġeometric function2 F 1 of matrix argument,useful in multivariate analysis

Summary LetS _i have the Wishart distributionW _p(∑_i,n_i) fori=1,2. An asymptotic expansion of the distribution of for large n=n₁+n₂ is derived, when∑ ₁∑ ₂ ⁻¹ =I+n^−1/2θ, based on an asymptotic solution of the system of partial differential equations for the hypergeometric function₂ F ₁, obtained recently by Muirhead [2]. Another asymptotic formula is also applied to the distributions of −2 log λ and −log|S ₂(S ₁+S ₂)⁻¹| under fixed∑ ₁∑ ₂ ⁻¹ , which gives the earlier results by Nagao [4]. Some useful asymptotic formulas for₁ F ₁ were investigated by Sugiura [7].     

Large sample nonparametric rejection of outlying observations

1. Summary Letx ₁≦, ..., ≦x _n be independent observations from continuous populations. The null hypothesis,H ₀, is that these observations are a sample. The alternative hypothesis is that thei smallest observations are too small (or that thei largest observations are too large) to be consistent withH ₀. Herei is a small number and should be specified without knowledge of the observation values. The common population hypothesized for the null case is assumed to be well-behaved but no specific assumptions are made about its shape. The alternative that thei smallest observations are too small is accepted if a statistic of the formx _i−(1+A)x _i+1+Ax _k is negative, whereA>0,k is the largest integer contained ini+√2n, andn is sufficiently large. Similarly, the alternative that thei largest observations are too large is accepted ifx _n+1−i −(1+A)x _n−i+Ax _n+1−k is positive. Two-sided tests are obtained as combinations of these one-sided tests. ForA suitably chosen, an approximate upper bound for the significance level of a test is evaluated from Techebycheff’s inequality. Using this relation, the value ofA is expressed as a function ofi, n, and the significance level upper bound. For fixed population shapes, these tests have powers that tend to unity for the case wheren−i of the populations are the same and the minimum difference between the median of this common population and the medians of the other populations increases in the appropriate direction The results of this paper may be useful in population statistics, operations research, and other applied fields.