Common nonnegative definite solutions of some classical matrix equations

A necessary and sufficient condition for the existence of a Hermitian nonnegative definite solution of system of matrix equations $$A_{1}X=C_{1},\qquad XB_{2}=C_{2}, \qquad A_{3}XA_{3}^{\ast}=C_{3},\qquad A_{4}XA_{4}^{\ast}=C_{4} $$ as well as a representation for this general nonnegative definite solution are derived. As particular cases, the corresponding results on some other systems are also derived.     

Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix

Let {X _k,i ; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let {p _n ; n ≥ 1} be a sequence of positive integers such that n/p _n is bounded away from 0 and ∞. In this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry ${L_{n}={\rm max}_{1\leq i < j\leq p_{n}}|\hat{\rho}^{(n)}_{i,j}|}$ of the sample correlation matrix ${{\bf {\Gamma}}_{n}=(\hat{\rho}_{i,j}^{(n)})_{1\leq i,j\leq p_{n}}}$ where ${\hat{\rho}^{(n)}_{i,j}}$ denotes the Pearson correlation coefficient between (X _1,i , ..., X _n,i )′ and (X _1,j ,...,X _n,j )′. Write ${F(x)= \mathbb{P}(|X_{1,1}|\leq x), x\geq0}$ , ${W_{c,n}={\rm max}_{1\leq i < j\leq p_{n}}|\sum_{k=1}^{n}(X_{k,i}-c)(X_{k,j}-c)|}$ , and ${W_{n}=W_{0,n},n\geq1,c\in(-\infty,\infty)}$ . Under the assumption that ${\mathbb{E}|X_{1,1}|^{2+\delta} < \infty}$ for some δ > 0, we show that the following six statements are equivalent: $$ {\bf (i)} \quad \lim_{n \to \infty} n^{2}\int\limits_{(n \log n)^{1/4}}^{\infty}\left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x}\right) \right) dF(x) = 0,$$ $$ {\bf (ii)}\quad n \mathbb{P}\left ( \max_{1 \leq i < j \leq n}|X_{1,i}X_{1,j} | \geq \sqrt{n \log n}\right ) \to 0 \quad{\rm as}\,n \to \infty,$$ $$ {\bf (iii)}\quad \frac{W_{\mu, n}}{\sqrt {n \log n}}\stackrel{\mathbb{P}}{\rightarrow} 2\sigma^{2},$$ $$ {\bf (iv)}\quad \left ( \frac{n}{\log n}\right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2,$$ $$ {\bf (v)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (\frac{W_{\mu, n}^{2}}{n \sigma^{4}} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8\pi}} e^{-t/2}\right \}, - \infty < t < \infty,$$ $$ {\bf (vi)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (n L_{n}^{2} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 \pi}} e^{-t/2}\right \}, - \infty < t < \infty$$ where ${\mu=\mathbb{E}X_{1,1}, \sigma^{2}=\mathbb{E}(X_{1,1} - \mu)^{2}}$ , and a _n  = 4 log p _n ? log log p _n . The equivalences between (i), (ii), (iii), and (v) assume that only ${\mathbb{E}X_{1,1}^{2} < \infty}$ . Weak laws of large numbers for W _n and L _n , n ≥  1, are also established and these are of the form ${W_{n}/n^{\alpha}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(\alpha > 1/2)$ and ${n^{1-\alpha}L_{n}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(1/2 < \alpha \leq 1)$ , respectively. The current work thus provides weak limit analogues of the strong limit theorems of Li and Rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of L _n obtained by Jiang. Some open problems are also posed.     

A Note on Drazin Invertibility for Upper Triangular Block Operators

A bounded linear operator A acting on a Banach space X is said to be an upper triangular block operators of order n, and we write ${A \in \mathcal{UT}_{n}(X)}$ , if there exists a decomposition of ${X = X_{1} \oplus . . . \oplus X_{n}}$ and an n × n matrix operator ${(A_{i,j})_{\rm 1 \leq i, j \leq n}}$ such that ${A = (A_{i, j})_{1 \leq i, j \leq n}, A_{i, j} = 0}$ for i > j. In this note we characterize a large set of entries A _{i, j} with j > i such that ${\sigma_{\rm D} (A) = {\bigcup\limits_{i = 1}^{n}} \sigma_{\rm D} (A_{i, i})}$ ; where σ_D(.) is the Drazin spectrum. Some applications concerning the Fredholm theory and meromorphic operators are given.     

Some properties of submatrices in a solution to the matrix equations AX=C, XB=D

Suppose that AX=C, XB=D has a common solution and partition its solution $X=\bigl[{\fontsize{7.5}{9}\selectfont \begin{array}{cc}X_{1}&X_{2}\\X_{3}&X_{4}\end{array}}\bigr]$ . In this paper, we give some formulas for the maximal and minimal ranks of the submatrices in a solution X to matrix equations AX=C, XB=D. In addition, we investigate the uniqueness and the independence of submatrices in a solutions X to this equations.     

Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

We consider the stochastic recursion ${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$ , where ${Q_n, X_n \in \mathbb{R}^d }$ , M _n are similarities of the Euclidean space ${ \mathbb{R}^d }$ and (Q _n , M _n ) are i.i.d. We study asymptotic properties at infinity of the invariant measure for the Markov chain X _n under assumption ${\mathbb{E}{[\log|M|]}=0}$ i.e. in the so called critical case.     

Small Divisor Problem in Dynamical Systems and Analytic Solutions of a q-Difference Equation with a Singularity at the Origin

In this paper, we consider a q-difference equation $$\sum_{j=0}^{k}\sum_{t=1}^{\infty}C_{t,j}(z)(y(q^jz))^{t}=G(z)$$ in the complex field ${\mathbb C,}$ where C _t,j (z) and G(z) have a h ₁ order pole and a h ₂ order pole at z = 0, respectively. Under the case 0 < |q| < 1 or |q| = 1, we give the existence of local analytic solutions for the above equation by using small divisor theory in dynamical systems.     

2-Reducible Two Paths and an Edge Constructing a Path in (2k + 1)-Edge-Connected Graphs

Let k ≥ 5 be an odd integer and G = (V(G), E(G)) be a k-edge-connected graph. For ${X\subseteq V(G),e(X)}$ denotes the number of edges between X and V(G) ? X. We here prove that if ${\{s_i,t_i\}\subseteq X_i\subseteq V(G)(i=1,2),f}$ is an edge between s ₁ and ${s_2,X_1\cap X_2=\emptyset,e(X_1)\le 2k-3,e(X_2)\le 2k-2}$ , and e(Y) ≥ k + 1 for each ${Y\subseteq V(G)}$ with ${Y\cap\{s_1,t_1,s_2,t_2\}=\{s_1,t_2\}}$ , then there exist paths P ₁ and P ₂ such that P _i joins s _i and ${t_i,V(P_i)\subseteq X_i}$ (i = 1, 2) and ${G-f-E(P_1\cup P_2)}$ is (k ? 2)-edge-connected, and in fact we give a generalization of this result.     

A system of matrix equations and its applications

We derive the solvability conditions and an expression of the general solution to the system of matrix equations A 1X=C1 , A2Y=C2 , YB2=D2 , Y=Y*, A3Z=C3 , ZB3=D3 , Z=Z*, B4X+(B4X)+C4YC4*+D4ZD4*=A4 . Moreover, we investigate the maximal and minimal ranks and inertias of Y and Z in the above system of matrix equations. As a special case of the results, we solve the problem proposed in Farid, Moslehian, Wang and Wu’s recent paper (Farid F O, Moslehian M S, Wang Q W, et al. On the Hermitian solutions to a system of adjointable operator equations. Linear Algebra Appl, 2012, 437: 1854-1891).     

Local probabilities for random walks conditioned to stay positive

Let S ₀ = 0, {S _n , n ≥ 1} be a random walk generated by a sequence of i.i.d. random variables X ₁, X ₂, . . . and let $\tau ^{-}={\rm min} \{ n \geq 1:S_{n}\leq 0 \}$ and $\tau ^{+}={\rm min}\{n\geq1:S_{n} > 0\} $ . Assuming that the distribution of X ₁ belongs to the domain of attraction of an α-stable law we study the asymptotic behavior, as ${n\rightarrow \infty }$ , of the local probabilities ${\bf P}{(\tau ^{\pm }=n)}$ and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities ${\bf P}{(S_{n} \in [x,x+\Delta )|\tau^{-} > n)}$ with fixed Δ and ${x=x(n)\in (0,\infty )}$ .     

A nonclassical LIL for sums of B-valued random variables when extreme terms are excluded

Let {X,X _n ; n≧1} be a sequence of B-valued i.i.d. random variables. Denote $X_{{n}}^{(r)}=X_{{m}}$ if ∥X _m ∥ is the r-th maximum of {∥X _k ∥; k≦n}, and let ${}^{(r)}S_{{n}}=S_{{n}}-(X_{{n}}^{(1)}+\cdots+X_{{n}}^{(r)})$ be the trimmed sums, where $S_{{n}}=\sum_{ k=1}^{n}X_{{k}}$ . Given a sequence of positive constants {h(n), n≧1}, which is monotonically approaching infinity and not asymptotically equivalent to loglogn, a limit result for $^{(r)}S_{{n}}/\sqrt{2nh(n)}$ is derived.     

On the Generalized Feynman–Kac Transformation for?Nearly Symmetric Markov Processes

Suppose that X is a right process which is associated with a non-symmetric Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on L ²(E;m). For $u\in D(\mathcal{E})$ , we have Fukushima??s decomposition: $\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$ . In this paper, we investigate the strong continuity of the generalized Feynman?CKac semigroup defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$ . Let $Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in D(\mathcal{E})_{b}$ . Denote by J ₁ the dissymmetric part of the jumping measure J of $(\mathcal{E},D(\mathcal{E}))$ . Under the assumption that J ₁ is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if and only if there exists a constant ?? ₀??0 such that $\|P^{u}_{t}\|_{2}\leq e^{\alpha_{0}t}$ for every t>0. If one of these conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on L ²(E;m). If X is equipped with a differential structure, then this result also holds without assuming that J ₁ is finite.     

Trivectors yielding spreads in PG(5, 2)

Given an alternating trilinear form ${T\in {\rm Alt}(\times^{3}V_{6})}$ on V ₆ = V(6, 2) let ${\mathcal{L}_{T}}$ denote the set of those lines ${\langle a, b \rangle}$ in ${{\rm PG}(5,2)=\mathbb{P}V_{6}}$ which are T-singular, satisfying, that is, T(a, b, x) = 0 for all ${x\in {\rm PG}(5, 2).}$ If ${\mathcal{L}_{21}}$ is a Desarguesian line-spread in PG(5, 2) it is shown that ${\mathcal{L}_{T}=\mathcal{L}_{21}}$ for precisely three choices T ₁,T ₂,T ₃ of T, which moreover satisfy T ₁ + T ₂ + T ₃ = 0. For ${T\in\mathcal{T}:=\{T_{1},T_{2},T_{3}\}}$ the ${\mathcal{G}_{T}}$ -orbits of flats in PG(5, 2) are determined, where ${\mathcal{G}_{T}\cong {\rm SL}(3,4).2}$ denotes the stabilizer of T under the action of GL(6, 2). Further, for a representative U of each ${\mathcal{G}_{T}}$ -orbit, the T-associate U ^# is also determined, where by definition $$U^{\#}=\{v\in {\rm PG}(5,2)\, |\, T(u_{1},u_{2},v) = 0\, \,{\rm for\,all }\, \, u_{1},u_{2}\in U\}$$ .     

On a certain exponential inequality for Gaussian processes

If X=(X _j ) _j=1 ^m is a zero-mean Gaussian stochastic process and $\sigma _{j}=\left( E{\big[} X_{j}^{2}{\big]} \right) ^{1/2},$ j=1,...,m, Tsirel’son (Theory Probab. Appl., 30, 820–828, 1985) and more explicitly Vitale (Ann. Probab., 24, 2172–2178, 1996 and A log-concavity proof for a Gaussian exponential bound. In: Hill, T.P., Houdré, C. (eds.) Advances in Stochastic Inequalities, Contemporary Mathematics, vol. 234, pp. 209–212. AMS, Providence, RI, 1999) applied results from Brunn–Minkowski theory to show that X satisfies the following inequality: $$ E\left[ \exp \left( \max_{1\leq j\leq m}{\bigg(}X_{j}-\frac{\sigma _{j}^{2}}{2} {\bigg)}\right) \right] \leq \exp \left( E\left[ \max_{1\leq j\leq m}X_{j}\right] \right). $$ In this paper a more general inequality will be derived using a known formula for Gaussian integrals. In particular, it also follows that $$ {\small \ }E\left[ \exp \left( \min_{1\leq j\leq m}{\bigg(}X_{j}-\frac{\sigma _{j}^{2}}{2}{\bigg)}\right) \right] \leq \exp \left( E\left[ \min_{1\leq j\leq m}X_{j}\right] \right) . $$ In the last section of this article the above exponential inequalities are combined with a well known variant of the Slepian lemma to compare certain option prices in the Black–Scholes and Bachelier models.     

A minimax theorem for vector-valued functions,part 2

We deal with the minimax problem relative to a vector-valued functionf: X ₀×Y ₀ »V, where a partial ordering in the topological vector spaceV is induced by a closed and convex coneC. In Ref. 1, under suitable hypotheses, we proved that $$Max\bigcup\limits_{s\varepsilon X_0 } {Min_w f(s,Y_0 )} \subset Min\bigcup\limits_{t\varepsilon Y_0 } {Maxf(X_0 ,t) + C;}$$ the exact meaning of the symbols is given in Section 2. In this work, we prove that, under a reasonable setting of hypotheses, the previous inclusion holds and also we have that $$Min_w \bigcup\limits_{t\varepsilon Y_0 } {Max} f(X_0 ,t) \subset Max\bigcup\limits_{s\varepsilon X_0 } {Min_w } f(s,Y_0 ) - C.$$      

An iterative algorithm for the least Frobenius norm Hermitian and generalized skew Hamiltonian solutions of the generalized coupled Sylvester-conjugate matrix equations

The iterative method of the generalized coupled Sylvester-conjugate matrix equations \(\sum\limits _{j=1}^{l}\left (A_{ij}X_{j}B_{ij}+C_{ij}\overline {X}_{j}D_{ij}\right )=E_{i} (i=1,2,\cdots ,s)\) over Hermitian and generalized skew Hamiltonian solution is presented. When these systems of matrix equations are consistent, for arbitrary initial Hermitian and generalized skew Hamiltonian matrices X _j (1), j = 1,2,? , l, the exact solutions can be obtained by iterative algorithm within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm Hermitian and generalized skew Hamiltonian solution of the problem. Finally, numerical examples are presented to demonstrate the proposed algorithm is efficient.     

Nash equilibria on topological semilattices

A function ${u : X \to \mathbb{R}}$ defined on a partially ordered set is quasi-Leontief if, for all ${x \in X}$ , the upper level set ${\{x\prime \in X : u(x\prime) \geq u(x)\}}$ has a smallest element; such an element is an efficient point of u. An abstract game ${u_{i} : \prod^{n}_{j=1} X_j \to \mathbb{R}, i \in \{1, \ldots , n\}}$ , is a quasi-Leontief game if, for all i and all ${(x_{j})_{j \neq i} \in \prod_{j \neq i} X_{j}, u_{i}((x_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ is quasi-Leontief; a Nash equilibrium x* of an abstract game ${u_{i} :\prod^{n}_{j=1} X_{j} \to \mathbb{R}}$ is efficient if, for all ${i, x^{*}_{i}}$ is an efficient point of the partial function ${u_{i}((x^{*}_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ . We establish the existence of efficient Nash equilibria when the strategy spaces X _i are topological semilattices which are Peano continua and Lawson semilattices.     

Schauder estimates for sub-elliptic equations

We establish Hölder estimates of second derivatives for a class of sub-elliptic partial differential operators in ${\mathbb{R}^{N}}$ of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the X _j ’s are smooth vector fields in ${\mathbb{R}^{N}}$ , and a _ij is a uniformly elliptic matrix. It is assumed that the X _j ’s satisfy homogeneity conditions with respect to a group of dilations δ _r which yield the existence of a composition law ${\circ}$ in ${\mathbb{R}^{N}}$ making the triplet ${\mathbb G=(\mathbb{R}^{N},\circ,\delta_{r})}$ an homogeneous Lie group on which the X _j ’s are left translation invariant. The Hölder norms are defined in terms of this composition law. The main tools used are the Taylor formula for smooth functions on ${\mathbb{G}}$ , some properties of the corresponding Taylor polynomials, and an orthogonality theorem that extends to homogeneous Lie groups a classical theorem of Calderón and Zygmund in the Euclidean setting.     

A perturbation result for semi-linear stochastic differential equations in UMD Banach spaces

We consider the effect of perturbations of A on the solution to the following semi-linear parabolic stochastic partial differential equation: $$\left\{\begin{array}{ll}{\rm d}U(t) & = AU(t)\,{\rm d}t + F(t,U(t))\,{\rm d}t + G(t,U(t))\,{\rm d}W_H(t), \quad t > 0;\\U(0)& = x_0. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad({\rm SDE})\end{array} \right.$$ Here, A is the generator of an analytic C ₀-semigroup on a UMD Banach space X, H is a Hilbert space, W _H is an H-cylindrical Brownian motion, ${G:[0,T]\times X\rightarrow \mathcal{L}(H, X_{\theta_G}^{A})}$ , and ${F : [0, T]\times X \rightarrow X_{\theta_F}^{A}}$ for some ${\theta_G > -\frac{1}{2}, \theta_F > -\frac{3}{2}+\frac{1}{\tau}}$ , where ${\tau\in [1, 2]}$ denotes the type of the Banach space and ${X_{\theta_F}^{A}}$ denotes the fractional domain space or extrapolation space corresponding to A. We assume F and G to satisfy certain global Lipschitz and linear growth conditions. Let A ₀ denote the perturbed operator and U ₀ the solution to (SDE) with A substituted by A ₀. We provide estimates for ${\|U - U_0\|_{L^p(\Omega;C([0,T];X))}}$ in terms of ${D_{\delta}(A, A_0) := \|R(\lambda : A) - R(\lambda : A_0)\|_{\mathcal{L}(X^{A}_{\delta-1},X)}}$ . Here, ${\delta\in [0, 1]}$ is assumed to satisfy ${0\leq \delta < {\rm min}\{\frac{3}{2} - \frac{1}{\tau} + \theta_F,\, \frac{1}{2} - \frac{1}{p} + \theta_G \}}$ . The work is inspired by the desire to prove convergence of space approximations of (SDE). In this article, we prove convergence rates for the case that A is approximated by its Yosida approximation.     

Circular law theorem for random Markov matrices

Let (X _jk ) _jk≥1 be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance σ ². Let M be the n?× n random Markov matrix with i.i.d. rows defined by ${M_{jk}=X_{jk}/(X_{j1}+\cdots+X_{jn})}$ . In particular, when X ₁₁ follows an exponential law, the random matrix M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Let λ₁, . . . , λ _n be the eigenvalues of ${\sqrt{n}M}$ i.e. the roots in ${\mathbb{C}}$ of its characteristic polynomial. Our main result states that with probability one, the counting probability measure ${\frac{1}{n}\delta_{\lambda_1}+\cdots+\frac{1}{n}\delta_{\lambda_n}}$ converges weakly as n→∞ to the uniform law on the disk ${\{z\in\mathbb{C}:|z|\leq m^{-1}\sigma\}}$ . The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent.