High-dimensional asymptotic expansions for the distributions of canonical correlations

This paper examines asymptotic distributions of the canonical correlations between and with q≤p, based on a sample of size of N=n+1. The asymptotic distributions of the canonical correlations have been studied extensively when the dimensions q and p are fixed and the sample size N tends toward infinity. However, these approximations worsen when q or p is large in comparison to N. To overcome this weakness, this paper first derives asymptotic distributions of the canonical correlations under a high-dimensional framework such that q is fixed, m=n−p→∞ and c=p/n→c₀∈[0,1), assuming that and have a joint (q+p)-variate normal distribution. An extended Fisher’s z-transformation is proposed. Then, the asymptotic distributions are improved further by deriving their asymptotic expansions. Numerical simulations revealed that our approximations are more accurate than the classical approximations for a large range of p,q, and n and the population canonical correlations.     

Alternating proximal algorithms for linearly constrained variational inequalities: Application to domain decomposition for PDE’s

Let X,Y,Z be real Hilbert spaces, let f:X→R∪{+∞}, g:Y→R∪{+∞} be closed convex functions and let A:X→Z, B:Y→Z be linear continuous operators. Let us consider the constrained minimization problem Given a sequence (γ_n) which tends toward 0 as n→+∞, we study the following alternating proximal algorithm where α and ν are positive parameters. It is shown that if the sequence (γ_n) tends moderately slowly toward 0, then the iterates of (A) weakly converge toward a solution of (P). The study is extended to the setting of maximal monotone operators, for which a general ergodic convergence result is obtained. Applications are given in the area of domain decomposition for PDE’s.     

On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification

Let be identically distributed random vectors in R^d, independently drawn according to some probability density. An observation is said to be a layered nearest neighbour (LNN) of a point if the hyperrectangle defined by and contains no other data points. We first establish consistency results on , the number of LNN of . Then, given a sample of independent identically distributed random vectors from R^d×R, one may estimate the regression function by the LNN estimate , defined as an average over the Y_i’s corresponding to those which are LNN of . Under mild conditions on r, we establish the consistency of towards 0 as n→∞, for almost all and all p≥1, and discuss the links between r_n and the random forest estimates of Breiman (2001) [8]. We finally show the universal consistency of the bagged (bootstrap-aggregated) nearest neighbour method for regression and classification.     

The Dirichlet Markov Ensemble

We equip the polytope of n×n Markov matrices with the normalized trace of the Lebesgue measure of Rⁿ². This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean (1/n,…,1/n). We show that if is such a random matrix, then the empirical distribution built from the singular values of tends as n→∞ to a Wigner quarter-circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of tends as n→∞ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of is of order when n is large.     

Singular value decomposition of large random matrices (for two-way classification of microarrays)

Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to random noise is investigated. It is proved that such an m×n random matrix almost surely has a constant number of large singular values (of order ), while the rest of the singular values are of order as m,n→∞. We prove almost sure properties for the corresponding isotropic subspaces and for noisy correspondence matrices. An algorithm, applicable to two-way classification of microarrays, is also given that finds the underlying block structure.     

Independence results for variants of sharply bounded induction

The theory , axiomatized by the induction scheme for sharply bounded formulae in Buss’ original language of bounded arithmetic (with ⌊x/2⌋ but not ⌊x/2^y⌋), has recently been unconditionally separated from full bounded arithmetic S₂. The method used to prove the separation is reminiscent of those known from the study of open induction.We make the connection to open induction explicit, showing that models of can be built using a “nonstandard variant” of Wilkie’s well-known technique for building models of IOpen. This makes it possible to transfer many results and methods from open to sharply bounded induction with relative ease.We provide two applications: (i) the Shepherdson model of IOpen can be embedded into a model of , which immediately implies some independence results for ; (ii) extended by an axiom which roughly states that every number has a least 1 bit in its binary notation, while significantly stronger than plain , does not prove the infinity of primes.     

On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices

Let X_n be n×N containing i.i.d. complex entries and unit variance (sum of variances of real and imaginary parts equals 1), σ>0 constant, and R_n an n×N random matrix independent of X_n. Assume, almost surely, as n→∞, the empirical distribution function (e.d.f.) of the eigenvalues of converges in distribution to a nonrandom probability distribution function (p.d.f.), and the ratio tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of converges in distribution. The limit is nonrandom and is characterized in terms of its Stieltjes transform, which satisfies a certain equation.     

Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings

In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces and Triebel–Lizorkin spaces for all s∈(0,1) and p,q∈(n/(n+s),∞], both in Rn and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve on Rn for all s∈(0,1) and q∈(n/(n+s),∞]. A metric measure space version of the above morphism property is also established.     

The limiting spectral distribution of the product of the Wigner matrix and a nonnegative definite matrix

Let W_n be n×n Hermitian whose entries on and above the diagonal are independent complex random variables satisfying the Lindeberg type condition. Let T_n be n×n nonnegative definitive and be independent of W_n. Assume that almost surely, as n→∞, the empirical distribution of the eigenvalues of T_n converges weakly to a non-random probability distribution.Let . Then with the aid of the Stieltjes transforms, we show that almost surely, as n→∞, the empirical distribution of the eigenvalues of A_n also converges weakly to a non-random probability distribution, a system of two equations determining the Stieltjes transform of the limiting distribution. Important analytic properties of this limiting spectral distribution are then derived by means of those equations. It is shown that the limiting spectral distribution is continuously differentiable everywhere on the real line except only at the origin and that a necessary and sufficient condition is available for determining its support. At the end, the density function of the limiting spectral distribution is calculated for two important cases of T_n, when T_n is a sample covariance matrix and when T_n is the inverse of a sample covariance matrix.     

Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings

Suppose that K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E. Let be two nonself asymptotically nonexpansive mappings with sequences {kn},{ln}⊂[1,∞), limn→∞kn=1, limn→∞ln=1, , respectively. Suppose {xn} is generated iteratively by

     

Densities for random balanced sampling

A random balanced sample (RBS) is a multivariate distribution with n components X_k, each uniformly distributed on [-1,1], such that the sum of these components is precisely 0. The corresponding vectors lie in an (n-1)-dimensional polytope M(n). We present new methods for the construction of such RBS via densities over M(n) and these apply for arbitrary n. While simple densities had been known previously for small values of n (namely 2,3, and 4), for larger n the known distributions with large support were fractal distributions (with fractal dimension asymptotic to n as n→∞). Applications of RBS distributions include sampling with antithetic coupling to reduce variance, and the isolation of nonlinearities. We also show that the previously known densities (for n?4) are in fact the only solutions in a natural and very large class of potential RBS densities. This finding clarifies the need for new methods, such as those presented here.     

The noncentral Wishart as an exponential family, and its moments

While the noncentral Wishart distribution is generally introduced as the distribution of the random symmetric matrix where Y₁,…,Y_n are independent Gaussian rows in with the same covariance, the present paper starts from a slightly more general definition, following the extension of the chi-square distribution to the gamma distribution. We denote by γ(p,a;σ) this general noncentral Wishart distribution: the real number p is called the shape parameter, the positive definite matrix σ of order k is called the shape parameter and the semi-positive definite matrix a of order k is such that the matrix ω=σaσ is called the noncentrality parameter. This paper considers three problems: the derivation of an explicit formula for the expectation of when Xγ(p,a,σ) and h₁,…,h_m are arbitrary symmetric matrices of order k, the estimation of the parameters (a,σ) by a method different from that of Alam and Mitra [K. Alam, A. Mitra, On estimated the scale and noncentrality matrices of a Wishart distribution, Sankhyā, Series B 52 (1990) 133–143] and the determination of the set of acceptable p’s as already done by Gindikin and Shanbag for the ordinary Wishart distribution γ(p,0,σ).     

The higher-order Picone identity and comparison of half-linear differential equations of even order

An identity of the Picone type for higher-order half-linear ordinary differential operators of the form and where p_j and P_j, j=0,…,n, are continuous functions defined on [a,b] and , is derived and then the Sturmian comparison theory for the corresponding 2nth-order equations l_α[x]=0 and L_α[y]=0 based on this identity is developed.     

Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing

Let be a fractional ARIMA(p,d,q) process with partial autocorrelation function α(·). In this paper, we prove that if d∈(−1/2,0) then |α(n)|∼|d|/n as n→∞. This extends the previous result for the case 0<d<1/2.     

Progressive Type II censored order statistics for multivariate observations

For a sequence of independent and identically distributed random vectors , i=1,2,…,n, we consider the conditional ordering of these random vectors with respect to the magnitudes of , where N is a p-variate continuous function defined on the support set of X₁ and satisfying certain regularity conditions. We also consider the Progressive Type II right censoring for multivariate observations using conditional ordering. The need for the conditional ordering of random vectors exists for example, in reliability analysis when a system has n independent components each consisting of p arbitrarily dependent and parallel connected elements. Let the vector of life lengths for the ith component of the system be , where denotes the life length of the jth element of the ith component. Then the first failure in the system occurs at time , and for this case . In this paper we introduce the conditionally ordered and Progressive Type II right-censored conditionally ordered statistics for multivariate observations and to study their distributional properties.     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.     

Asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis when the dimension is large

This paper examines asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis based on a sample of size N=n+1 on two sets of variables, i.e.,  and . These problems are related to dimension reduction. The asymptotic approximations of the statistics have been studied extensively when dimensions p₁ and p₂ are fixed and the sample size N tends to infinity. However, the approximations worsen as p₁ and p₂ increase. This paper derives asymptotic expansions of the test statistics when both the sample size and dimension are large, assuming that and have a joint (p₁+p₂)-variate normal distribution. Numerical simulations revealed that this approximation is more accurate than the classical approximation as the dimension increases.