Some limit theorems on the eigenvectors of large dimensional sample covariance matrices

Let {v_ij} i,j = 1, 2,…, be i.i.d. standardized random variables. For each n, let V_n = (v_ij) i = 1, 2,…, n; j = 1, 2,…, s = s(n), where $\text{(}\text{n}\text{s}\text{) → y > 0}$ as n → ∞, and let $\text{M}{}_{\mathrm{n}}\text{= (}\text{1}\text{s}\text{)V}{}_{\mathrm{n}}\text{V}{}_{\mathrm{n}}{}^{\mathrm{T}}$. Previous results [7, 8] have shown the eigenvectors of M_n to display behavior, for n large, similar to those of the corresponding Wishart matrix. A certain stochastic process X_n on [0, 1], constructed from the eigenvectors of M_n, is known to converge weakly, as n → ∞, on D[0, 1] to Brownian bridge when v₁₁ is N(0, 1), but it is not known whether this property holds for any other distribution. The present paper provides evidence that this property may hold in the non-Wishart case in the form of limit theorems on the convergence in distribution of random variables constructed from integrating analytic function w.r.t. X_n(F_n(x)), where F_n is the empirical distribution function of the eigenvalues of M_n. The theorems assume certain conditions on the moments of v₁₁ including E(v₁₁⁴) = 3, the latter being necessary for the theorems to hold.     

Duality theorems for perturbed convex optimization

We show that if F, X are two locally convex spaces and $\text{h: F →}\text{R}\text{?}\text{, ?: F × X → R}$ are two convex functionals satisfying h(y) = ?(y, x₀) (y?F) for some x₀?X, then, under suitable assumptions, the computation of inf h(F) can be reduced to the computation of inf ?(H) on certain hyperplanes H of F × X. We give some applications.     

Duality theorems for linear systems and convex systems

We show that, if (F →^uX) is a linear system, $\text{Ω ? X}$ a convex target set and $\text{h: X →}\text{R}\text{?}$ a convex functional, then, under suitable assumptions, the computation of inf $\text{h({y ? F ¦ u(y) ? Ω}}$) can be reduced to the computation of the infimum of h on certain strips or hyperplanes in F, determined by elements of $\text{u}{}^1\text{(X}{}^1\text{)}$, or of the infima on F of Lagrangians, involving elements of $\text{u}{}^1\text{(X}{}^1\text{)}$. Also, we prove similar results for a convex system (F →^uX) and the convex cone Ω of all non-positive elements in X.     

Diffusion approximations of age-and-position dependent branching processes

By a (G, F, h) age-and-position dependent branching process we mean a process in which individuals reproduce according to an age dependent branching process with age distribution function G(t) and offspring distribution generating function F, the individuals (located in R^N) can not move and the distance of a new individual from its parent is governed by a probability density function h(r). For each positive integer n, let Z_n(t,dx) be the number of individuals in dx at time t of the (G, F_n,h_n) age-and-position dependent branching process. It is shown that under appropriate conditions on G, F_n and h_n, the finite dimensional distribution of $\text{Z}{}_{\mathrm{n}}\text{(nt, dx)}\text{n}$ converges, as n → ∞, to the corresponding law of a diffusion continuous state branching process X(t,dx) determined by a ψ-semigroup {ψ_t: t ? 0}. The ψ-semigroup {ψ_t} is the solution of a non-linear evolution equation. A semigroup convergence theorem due to Kurtz [10], which gives conditions for convergence in distribution of a sequence of non-Markovian processes to a Markov process, provides the main tools.     

A remark on the tail probability of a distribution

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let S_n = Σ_j=1ⁿX_j and $\text{E{N}{}_{\mathrm{\infty }}\text{(r, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{r?2}}\text{P{|S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>r</mtext><mtext>t</mtext>}}\text{}}$. In this paper, we prove that (1) lim_?→0+?^α(r?1)E{N_∞(r, t, ?)} = K(r, t) if E(X₁) = 0, Var(X₁) = 1, and E(| X₁ |^t) < ∞, where 2 ≤ t < 2r ≤ 2t, $\text{K(r, t) =}\text{{2}{}^{\mathrm{<mtext>\alpha (r?1)</mtext><mtext>2</mtext>}}\text{Γ(}\text{(1 + α(r ? 1))}\text{2}\text{)}}\text{{(r ? 1) Γ(}\text{1}\text{2}\text{)}}$, and $\text{α =}\text{2t}\text{(2r ? t)}$; (2) $\text{lim}{}_{\mathrm{?\to 0+}}\text{G(t, ?)}\text{H(t, ?)}\text{= 0}$ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(|X₁|^t) < ∞, where G(t, ?) = E{N_∞(t, t, ?)} = Σ_n=1^∞n^t?2P{| S_n | > ?n} → ∞ as ? → 0+ and $\text{H(t, ?) = E{N}{}_{\mathrm{\infty }}\text{(t, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{t?2}}\text{P{| S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>2</mtext><mtext>t</mtext>}}\text{} → ∞}\text{as}\text{? → 0+}$, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(| X₁ |^t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.     

On Berry-Esséen rates,a law of the iterated logarithm and an invariance principle for the proportion of the sample below the sample mean

Let F_n(x) be the empirical distribution function based on n independent random variables X₁,…,X_n from a common distribution function F(x), and let $\text{X}\text{= Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{X}{}_{\mathrm{i}}\text{n}$ be the sample mean. We derive the rate of convergence of $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$ to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$.     

On admissible shifts of generalized white noises

Let $\text{S}$ be the Schwartz space of rapidly decreasing real functions. The dual space $\text{S}$¹ consists of the tempered distributions and the relation $\text{S}$ ? L²($\text{R}$) ? $\text{S}$¹ holds. Let γ be the Gaussian white noise on $\text{S}$¹ with the characteristic functional $\text{γ}\text{(ξ) =}\text{exp}\text{{?∥ξ∥}{}^2\text{/2}}$, ξ ∈ $\text{S}$, where ∥·∥ is the L²($\text{R}$)-norm. Let ν be the Poisson white noise on $\text{S}$¹ with the characteristic functional $\text{ν}\text{(ξ)}$ = exp?_$\text{R}$∫_$\text{R}$ {[exp(iξ(t)u)] ? 1 ? (1 + u²)^?1(iξ(t)u)} dη(u)dt), ξ ∈ $\text{S}$, where the Lévy measure is assumed to satisfy the condition ∫_$\text{R}$u²dη(u) < ∞. It is proved that γ1ν has the same dichotomy property for shifts as the Gaussian white noise, i.e., for any ω ∈ $\text{S}$¹, the shift $\text{(γ1ν)}{}_{\mathrm{\omega }}$ of γ1ν by ω and γ1ν are either equivalent or orthogonal. They are equivalent if and only if when ω ∈ L²($\text{R}$) and the Radon-Nikodym derivative is derived. It is also proved that for the Poisson white noice ν_ω is orthogonal to ν for any non-zero ω in $\text{S}$¹.     

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.     

The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for ∝ λ (dx)6Pⁿ(x, ·) - π6 to be of order o(ψ(n)^-1), where Pⁿ (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, π is the invariant probability measure, λ an initial distribution and ψ belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order ψ, which in a countable state space is equivalent to $\text{Σ ψ(n)}\text{P}{}_{\mathrm{i}}\text{{τ}{}_{\mathrm{i}}\text{?n} <∞}$ for some i, where τ_i is the hitting time of the tate i. We also show that for a general Markov chain to be ergodic of order ψ it suffices that a corresponding condition is satisfied by a small set.We apply these results to non-singular renewal measures on $\text{R}$ providing a probabilisite method to estimate the right tail of the renewal measure when the increment distribution F satisfies ∝ tF(dt) 0; > 0 and ∝ ψ(t)(1- F(t))dt< ∞.     

Some improvements in Neuberger's iteration procedure for solving partial differential equations

If m and n are positive integers then let S(m, n) denote the linear space over R whose elements are the real-valued symmetric n-linear functions on E_m with operations defined in the usual way. If $\text{U}$ is a function from some sphere S in E_m to R then let $\text{U}$⁽ⁱ⁾(x) denote the ith Frechet derivative of $\text{U}$ at x. In general $\text{U}$⁽ⁱ⁾(x)∈S(m,i). In the paper “An Iterative Method for Solving Nonlinear Partial Differential Equations” [Advances in Math. 19 (1976), 245–265] Neuberger presents an iterative procedure for solving a partial differential equation of the form

$\text{AU}{}^{\mathrm{n}}\text{(x)=F(x, U(x), U′(x),…,U}{}^{\mathrm{k}}\text{(x))}$

, where k > n, $\text{U}$ is the unknown from some sphere in E_m to R, A is a linear functional on S(m, n), and F is analytic. The defect in the theory presented there was that in order to prove that the iterates converged to a solution $\text{U}$ the condition $\text{k ?}\text{n}\text{2}$ was needed. In this paper an iteration procedure that is a slight variation on Neuberger's procedure is considered. Although the condition $\text{k ?}\text{n}\text{2}$ cannot as yet be eliminated, it is shown that in a broad class of cases depending upon the nature of the functional A the restriction $\text{k ?}\text{n}\text{2}$ may be replaced by the restriction $\text{k ?}\text{3n}\text{4}$.     

Retraction spaces and the homotopy metric

Let X be a finite-dimensional compactum. Let $\text{R}$(X) and $\text{N}$(X) be the spaces of retractions and non-deformation retractions of X, respectively, with the compact-open (=sup-metric) topology. Let 2^X_h be the space of non-empty compact ANR subsets of X with topology induced by the homotopy metric. Let R^X_h be the subspace of 2^X_h consisting of the ANR's in X that are retracts of X.We show that $\text{N}$(S^m) is simply-connected for m > 1. We show that if X is an ANR and A₀?R^X_h, then lim_i→∞A_i=A₀ in 2^X_h if and only if for every retraction r₀ of X onto A₀ there are, for almost all i, retractions r_i of X onto A_i such that lim_i→∞r_i=r_o in $\text{R}$(X). We show that if X is an ANR, then the local connectedness of $\text{R}$(X) implies that of R^X_h. We prove that $\text{R}$(M) is locally connected if M is a closed surface. We give examples to show how some of our results weaken when X is not assumed to be an ANR.     

Three problems on the lengths of increasing runs

Let U₁, U₂,… be a sequence of independent, uniform (0, 1) r.v.'s and let R₁, R₂,… be the lengths of increasing runs of {U_i}, i.e., X₁=R₁=inf{i:U_i+1<U_i},…, X_n=R₁+R₂+?+R_n=inf{i:i>X_n?1,U_i+1<U_i}. The first theorem states that the sequence $\text{(}\text{3}\text{2}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(X}{}_{\mathrm{n}}\text{?2n)}$ can be approximated by a Wiener process in strong sense.Let τ(n) be the largest integer for which R₁+R₂+?+R_τ(n)?n, $\text{R}{}^1{}_{\mathrm{n}}\text{=n?(R}{}_1\text{+R}{}_2\text{+?+R}{}_{\mathrm{\tau (n)}}\text{)}$ and $\text{M}{}_{\mathrm{n}}\text{=}\text{max}\text{{R}{}_1\text{,R}{}_2\text{,…,R}{}_{\mathrm{\tau (n)}}\text{,R}{}^1{}_{\mathrm{n}}\text{}}$. Here M_n is the length of the longest increasing block. A strong theorem is given to characterize the limit behaviour of M_n.The limit distribution of the lengths of increasing runs is our third problem.     

Central limit theorems for C(S)-valued random variables

Let C(S) be the space of real-valued continuous functions on a compact metric space S. Let {X_n, n ? 1} be a sequence of independent identically distributed C(S)-valued random variables with mean zero and sup_t?sE[X₁²(t)] = 1. We show that the measures induced by ($\text{X}{}_1\text{+ ··· + X}{}_{\mathrm{n}}\text{) n}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ converge weakly to a Gaussian measure on C(S) under different conditions on X₁, one of which consolidates and extends results of Strassen and Dudley, Giné, and Dudley. Our method of proof is different from the methods employed by these authors.     

A central limit theorem for the sojourn times of strongly ergodic Markov chains

Let X_n be an irreducible aperiodic recurrent Markov chain with countable state space I and with the mean recurrence times having second moments. There is proved a global central limit theorem for the properly normalized sojourn times. More precisely, if $\text{t}{}^{\mathrm{(n}}\text{)}{}_{\mathrm{i}}\text{=Σ}{}^{\mathrm{n}}{}_{\mathrm{k=1}}\text{i}\text{?}{}_{\mathrm{i}}\text{(X}{}_{\mathrm{k}}\text{)}$, then the probability measures induced by {t⁽ⁿ⁾_i/√n?√nπ_i}_i?I(π_i being the ergotic distribution) on the Hilbert-space of square summable I-sequences converge weakly in this space to a Gaussian measure determined by a certain weak potential operator.     

A general theory of surrogate dual and perturbational extended surrogate dual optimization problems

For the optimization problem (P) α = inf h(G), where G ≠ Ø is a subset of a locally convex space $\text{F}\text{and}\text{h: F →}\text{R}\text{?}$, we introduce and study two general concepts of dual problems, encompassing the classical surrogate dual problem. The first one involves only a family of surrogate constraints sets Δ_{G, Φ} ? F (Φ ∈ W), where W ? R^X, X being a locally convex space. The second one uses a perturbation functional $\text{?: F × X →}\text{R}\text{?}$ and a family of sets $\text{}\text{?}\text{gD}{}_{\mathrm{(F,x0),\Phi }}\text{? F × X (Φ ∈ W)}$, where W ? R^X. We give duality theorems, introduce Lagrangians, and show some relations between these problems and the dual problems to (P) defined with the aid of a perturbation and a concept of conjugation of functionals.     

Reflection classes whose cubes cover the alternating group

A reflection class (REC) over a finite set A is a conjugacy class of a reflection (permutation of order ? 2) of A. It was known that for no REC X, X² = Alt(n) holds, and that for some RECs X, X⁴ = Alt(n) holds (n ? 5). Let i > 0, and let c(θ) denote the number of cycles of θ?S(n). Let X_i = {ψ ∈ S(n): ψ² = 1, ψ has exactly i fixed points}. We prove that θ?X_i³ if and only if: (1) i ≡ n (mod 2); (2) The parity of X_i equals the parity of θ; and (3) $\text{i ?}\text{1}\text{3}\text{(n + 2 c(θ))}$. As a consequence, {X: X is a REC, X³ = Alt(n)} and {X: X is a REC, X³ = S(n) ? Alt(n)} are determined.     

Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

Modes of convergence of Markov chain transition probabilities

Suppose {Pⁿ(x, A)} denotes the transition law of a general state space Markov chain {X_n}. We find conditions under which weak convergence of {X_n} to a random variable X with law L (essentially defined by ∝ Pⁿ(x, dy) g(y) → ∝ L(dy) g(y) for bounded continuous g) implies that {X_n} tends to X in total variation (in the sense that $\text{∥ P}{}^{\mathrm{n}}\text{(x,}{}^{\mathrm{.}}\text{) ? L ∥ → 0)}$, which then shows that L is an invariant measure for {X_n}. The conditions we find involve some irreducibility assumptions on {X_n} and some continuity conditions on the one-step transition law {P(x, A)}.