Asymptotic distributions of functions of the eigenvalues of sample covariance matrix and canonical correlation matrix in multivariate time series

Let S = (1/n) Σ_t=1ⁿ X(t) X(t)′, where X(1), …, X(n) are p × 1 random vectors with mean zero. When X(t) (t = 1, …, n) are independently and identically distributed (i.i.d.) as multivariate normal with mean vector 0 and covariance matrix Σ, many authors have investigated the asymptotic expansions for the distributions of various functions of the eigenvalues of S. In this paper, we will extend the above results to the case when {X(t)} is a Gaussian stationary process. Also we shall derive the asymptotic expansions for certain functions of the sample canonical correlations in multivariate time series. Applications of some of the results in signal processing are also discussed.     

Self-adjusting stochastic processes

Let A_t(i, j) be the transition matrix at time t of a process with n states. Such a process may be called self-adjusting if the occurrence of the transition from state h to state k at time t results in a change in the hth row such that A_t+1(h, k) ? A_t(h, k). If the self-adjustment (due to transition h → kx) is A_{t + 1}(h, j) = λA_t(h, j) + (1 ? λ)δ_jk (0 < λ < 1), then with probability 1 the process is eventually periodic. If A₀(i, j) < 1 for all i, j and if the self-adjustment satisfies A_{t + 1}(h, k) = ?(A_t(h, k)) with ?(x) twice differentiable and increasing, x < ?(x) < 1 for 0 ? x < 1,?(1) = ?′(1) = 1, then, with probability 1, lim A_t does not exist.     

On the rate for uniform strong consistency of empirical distributions of independent nonidentically distributed multivariate random variables

Let X_j = (X_1j ,…, X_pj), j = 1,…, n be n independent random vectors. For x = (x₁ ,…, x_p) in R^p and for α in [0, 1], let F_j${}^1$(x) = αI(X_1j < x₁ ,…, X_pj < x_p) + (1 ? α) I(X_1j ≤ x₁ ,…, X_pj ≤ x_p), where I(A) is the indicator random variable of the event A. Let F_j(x) = E(F_j${}^1$(x)) and D_n = sup_{x, α} max_{1 ≤ N ≤ n} |Σ₀ⁿ(F_j${}^1$(x) ? F_j(x))|. It is shown that P[D_n ≥ L] < 4pL exp{?2(L²n^?1 ? 1)} for each positive integer n and for all L² ≥ n; and, as n → ∞, $\text{D}{}_{\mathrm{n}}\text{= 0((n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ with probability one.     

A new result on Davenport constant

Let G be a finite abelian group of order n and Davenport constant D(G). Let S=⁰h(S)_∏g∈Ggvg(S)∈F(G) be a sequence with a maximal multiplicity h(S) attained by 0 and t=|S|?n+D(G)−1. Then 0∈k_∑(S) for every 1?k?t+1−D(G). This is a refinement of the fundamental result of Gao [W.D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996) 100-103].     

Accurate approximation of time-varying matrices of linear differential systems

We consider the problem of the identification of the time-varying matrix A(t) of a linear m-dimensional differential system y′ = A(t)y. We develop an approximation A_n,k = ∑ⁿ_{j ? 1}c_j{Y(t_k + τ_j) Y^?1(t_k) ? I} to A(t_k) for grid points t_k = a + kh, k = 0,…, N using specified τ_j = θ_jh, 0 < θ_j < 1, j = 1, …, n, and show that for each t_k, the L₁ norm of the error matrix is $\text{O}$(hⁿ). We demonstrate an efficient scheme for the evaluation of A_n,k and treat sample problems.     

Some well-posed functional equations which generate semigroups

Consider the abstract linear functional equation (FE) (Dx)(t) = f(t) (t ? 0), x(t) = ?(t) (t ? 0) in a Banach space B. A theorem is proven which contains the following result as a special case. Let Y(R; B; η) be a L^p-space or C₀-space on R = (?t8, ∞), with a suitable weight function η, and with values in B. Let D be a closed (unbounded) causal linear operator in Y(R; B; η), which commutes with translations. Suppose that D + λI has a continuous causal inverse for some complex λ, and that D restricted to those functions in Y(R;B;η) which vanish on R^? = (?∞, 0] has a continuous causal inverse. Then (FE) generates a strongly continuous semigroup of translation type on a Banach space, which is essentially the cross product of the restriction of the domain of D to R^? and Y(R⁺; B; η). Examples with B = Cⁿ on how the theory applies to a neutral functional differential equation, a difference equation, a Volterra integrodifferential equation (with nonintegrable kernel but integrable resolvent), and a fractional order functional differential equation are given. Also, an abstract neutral functional differential equation in a Hilbert space is studied and applications to an abstract Volterra integrodifferential equation in a Banach space are indicated.     

Fundamental solution for linear two-point boundary value problem

The quadratic functional minimization with differential restrictions represented by the command linear systems is considered. The optimal solution determination implies the solving of a linear problem with two points boundary values. The proposed method implies the construction of a fundamental solution S(t)—a n×n matrix—and of a vector h(t) defining an adjoint variable λ(t) depending of the state variable x(t). From the extremum necessary conditions it is obtained the Riccati matrix differential equation having the S(t) as unknown fundamental solution is obtained. The paper analyzes the existence of the Riccati equation solution S(t) and establishes as the optimal solution of the proposed optimum problem. Also a superior limit of the minimum for the considered quadratic functionals class are evaluated.     

A characterization of the multivariate normal distribution

Let X_j (j = 1,…,n) be i.i.d. random variables, and let Y′ = (Y₁,…,Y_m) and X′ = (X₁,…,X_n) be independently distributed, and A = (a_jk) be an n × n random coefficient matrix with a_jk = a_jk(Y) for j, k = 1,…,n. Consider the equation U = AX, Kingman and Graybill [Ann. Math. Statist.41 (1970)] have shown U ～ N(O,I) if and only if X ～ N(O,I). provided that certain conditions defined in terms of the a_jk are satisfied. The task of this paper is to delete the identical assumption on X₁,…,X_n and then generalize the results to the vector case. Furthermore, the condition of independence on the random components within each vector is relaxed, and also the question raised by the above authors is answered.     

Asymptotic enumeration of dense 0-1 matrices with specified line sums

Let s=(s₁,s₂,…,sm) and t=(t₁,t₂,…,tn) be vectors of non-negative integers with . Let B(s,t) be the number of m×n matrices over {0,1} with jth row sum equal to sj for 1?j?m and kth column sum equal to tk for 1?k?n. Equivalently, B(s,t) is the number of bipartite graphs with m vertices in one part with degrees given by s, and n vertices in the other part with degrees given by t. Most research on the asymptotics of B(s,t) has focused on the sparse case, where the best result is that of Greenhill, McKay and Wang (2006). In the case of dense matrices, the only precise result is for the case of equal row sums and equal column sums (Canfield and McKay, 2005). This paper extends the analytic methods used by the latter paper to the case where the row and column sums can vary within certain limits. Interestingly, the result can be expressed by the same formula which holds in the sparse case.     

R/S统计量重对数律的精确收敛速度

Let{Xn;n≥1}be a sequence of i.i.d, random variables with finite variance,Q（n）be the related R/S statistics. It is proved that lim ε↓0 ε^2 ∑n=1 ^8 n log n/1 P{Q（n）≥ε√2n log log n}=2/1 EY^2,where Y=sup0≤t≤1B（t）-inf0≤t≤sB（t）,and B（t） is a Brownian bridge.     

A Fisher type inequality

In this paper we study subsets of a finite set that intersect each other in at most one element. Each subset intersects most of the other subsets in exactly one element. The following theorem is one of our main conclusions. Let S₁,… S_m be m subsets of an n-set S with |S₁| ? 2 (l = 1, …,m) and |S_i ∩ S_j| ? 1 (i ≠ j; i, j = 1, …, m). Suppose further that for some fixed positive integer c each S_i has non-empty intersection with at least m ? c of the remaining subsets. Then there is a least positive integer M(c) depending only on c such that either m ? n or m ? M(c).     

An asymptotic formula for the maximum size of an h-family in products of partially ordered sets

An h-family of a partially ordered set P is a subset of P such that no h + 1 elements of the h-family lie on any single chain. Let S₁, S₂,… be a sequence of partially ordered sets which are not antichains and have cardinality less than a given finite value. Let P_n be the direct product of S₁,…, S_n. An asymptotic formula of the maximum size of an h-family in P_n is given, where $\text{h=o(}\text{n}\text{)}$ and n → ∞.     

Characterization of the absence of a spectral gapfor a class of periodic Jacobi operators

Let q ∈ {2, 3} and let 0 = s₀ < s₁ < … < s_q = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set l_j = s_j − s_j−1 for j ∈ 1, q. Given (p₁,, p_q) ∈ R^q, let b: Z → R be a periodic function of period T such that b(·) = p_j on s_j−1 + 1, s_j for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l²(Z). By [λ_2j , λ_2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that p_m ≠ p_n for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1.     

Partitions of large multipartites with congruence conditions. II

For 1 ≦ l ≦ j, let $\text{a}$_l = ?_h=1^q(l){a_lh + Mv: v = 0, 1, 2,…}, where j, M, q(l) and the a_lh are positive integers such that j > 1, a_l1 < … < a_lq(2) ≦ M, and let $\text{a}$_l^′ = $\text{a}$_l ∪ {0}. Let p(n : $\text{B}$) be the number of partitions of n = (n₁,…,n_j) where, for 1 ≦ l ≦ j, the lth component of each part belongs to $\text{B}$_l and let p¹(n : $\text{B}$) be the number of partitions of n into different parts where again the lth component of each part belongs to $\text{B}$_l. Asymptotic formulas are obtained for p(n : $\text{a}$), p¹(n : $\text{a}$) where all but one n_l tend to infinity much more rapidly than that n_l, and asymptotic formulas are also obtained for p(n : $\text{a}$′), p¹(n ; $\text{a}$′), where one n_l tends to infinity much more rapidly than every other n_l. These formulas contrast with those of a recent paper (Robertson and Spencer, Trans. Amer. Math. Soc., to appear) in which all the n_l tend to infinity at approximately the same rate.     

Равносходимость разложений в кратный ряд и интеграл фурье, «прямоугольные частичные суммы» которых рассматриваются по некоторой подпоследовательности

In this paper we investigate the problem of the equiconvergence on T ^N = [-π, π) ^N of the expansions in multiple trigonometric series and Fourier integral of functions f ∈ L _p (T ^N ) and g ∈ L _p (? ^N ), where p > 1, N ≥ 3, g(x) = f(x) on T ^N , in the case when the “rectangular partial sums” of the indicated expansions, i.e.,– _n (x; f) and J _α(x; g), respectively, have indices n ∈ ? ^N and α ∈ ? ^N (n _j = [α _j ], j = 1,...,N, [t] is the integer part of t ∈ ?¹), in those certain components are the elements of “lacunary sequences”.     

On the functional equationS(m,n)F[n/(m,n)] =F(n)h[n/(m,n)]

In this paper, we discuss the pairs (f, h) of arithmetical functions satisfying the functional equation in the title, whereF is the product off andh under the Dirichlet convolution; that is,F(n) = Σ _d|n ?(d)h(n/d) andS(m n) = Σd|(m, n) ?(d)h(n/d). The well-known Hölder's identity is a special case of this functional equation (?(n) =n, h(n) = μ(n)). We also generalize the functional equation in the title to any arbitrary regular arithmetical convolution and discuss the pairs of solutions (f, h) of the generalized functional equation and pose some problems relating to the characterization of all pairs of solutions.     

Simultaneous congruence of convex compact sets of Hermitian matrices with constant rank

Let S be a compact convex set of n × n hermitian matrices (n ⩾ 2). Suppose every member of S is nonsingular and has exactly one negative eigenvalue. Let (ε₁,…,ε_n) be any ordered n-tuple from the set {- 1, 1}. One of our main results is that a nonsingular matrix X exists such that, for every A in S and every 1 ⩽ j ⩽ n, the (j, j) entry of X¹AX has sign ε_j. A similar result, with only negative ε_j allowed, is proved also for a compact convex set S of n × n hermitian matrices such that every member of S has the same rank and exactly one negative eigenvalue.     

On the h-triangles of sequentially (S r ) simplicial complexes via algebraic shifting

Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially (S _r ) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen–Macaulay and satisfying Serre’s condition (S _r ). Let Δ be a (d?1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let (h _i,j (Δ))_{0≤j≤i≤d} be the h-triangle of Δ and (h _i,j (Γ(Δ)))_{0≤j≤i≤d} be the h-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially (S _r ) and for every i and j with 0≤j≤i≤r?1, the equality h _i,j (Δ)=h _i,j (Γ(Δ)) holds true.     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

The characterization of sort sequences

A sort sequenceS _n is a sequence of all unordered pairs of indices inI _n = 1, 2, ...,n. With a sort sequenceS _n, we associate a sorting algorithmA(S _n) to sort input setX = x ₁, x₂, ..., x_n as follows. An execution of the algorithm performs pairwise comparisons of elements in the input setX as defined by the sort sequenceS _n, except that the comparisons whose outcomes can be inferred from the outcomes of the previous comparisons are not performed. Let χ(S _n) denote the average number of comparisons required by the algorithmA(S _n assuming all input orderings are equally likely. Let χ*(n) and χ°(n) denote the minimum and maximum values, respectively, of χ(S _n) over all sort sequencesS _n. Exact determination of χ*(n), χO(n) and associated extremal sort sequences seems difficult. Here, we obtain bounds on χ*(n) and χO(n).