Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

Common strict character of some sharp infinite-sequence martingale inequalities

A procedure is given for proving strictness of some sharp, infinite-sequence martingale inequalities, which arise from sharp, finite-sequence martingale inequalities attained by degenerating extremal distributions. The procedure is applied to obtain strictness of the sharp inequalities of Cox and Kemperman

$\text{P(|X}{}_{\mathrm{i}}\text{|?1 for some i=1, 2,…)?(ln 2)}{}^{\mathrm{?1}}\text{sup}\text{n}\text{E}\text{∑}\text{i=0}\text{n}\text{X}{}_{\mathrm{i}}$

and of Cox (sharp form of Burkholder's inequality)

$\text{P}\text{∑}\text{i=0}\text{∞}\text{X}{}^2{}_{\mathrm{i}}\text{?1}\text{? e}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{sup}\text{n}\text{E}\text{∑}\text{i=0}\text{n}\text{X}{}_{\mathrm{i}}$

for all nontrivial martingale difference sequences X₀,X₁,….     

Orthogonal polynomials on the negative multinomial distribution

Orthogonal polynomials on the multivariate negative binomial distribution,

$\text{(1 + Θ)}{}^{\mathrm{?\alpha ?x}}\text{(}\text{π}\text{j=0}\text{p}\text{Θ}{}_{\mathrm{j}}{}^{\mathrm{x}}\text{j}\text{x}{}_{\mathrm{j}}\text{!}\text{)}\text{Γ(α + x)}\text{Γ(α)}$

where α > 0, Θ₁ > 0, x = ΣΘ_i, x₀, x₁, …, x_p = 0,1, … are constructed and their properties studied.     

A complete set of invariants for finite groups and other results

For a finite group G and a set I ? {1, 2,…, n} let

${}_{\mathrm{G}}\text{(n,I) = ∑}{}_{\mathrm{g\; \in \; G}}\text{ε}{}_1\text{(g)?ε}{}_2\text{(g)???ε}{}_{\mathrm{n}}\text{(g)}$

,where

$\text{ε}{}_{\mathrm{i}}\text{(g)=g}\text{if}\text{i=∈ I,}$

$\text{ε}{}_{\mathrm{l}}\text{(g)=l}\text{if}\text{i=∈ I.}$

We prove, among other results, that the positive integers

$\text{tr}\text{(e}{}_{\mathrm{G}}\text{(n,I}{}_1\text{)+?+e}{}_{\mathrm{G}}\text{(n,I}{}_{\mathrm{r}}\text{))}{}^{\mathrm{k}}\text{:n,r,k,?1, I}{}_{\mathrm{j}}\text{?{1,…,n}, 1?|i}{}_{\mathrm{j}}\text{|?3}$

for 1 ? j ? r, I_j1 ∩ I_j2 ∩ I_j3 ∩ I_j4 = Ø for any 1 ? j₁ <j₂ <j₃ <j₄ ? r, determine G up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups.     

Strong approximation theorems for density dependent Markov chains

A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form

$\text{X}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lY}{}_1\text{N ∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X}{}_{\mathrm{N}}\text{(s))ds}$

where $\text{l∈}\text{Z}{}^{\mathrm{t}}$, the Y₁ are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution).The corresponding deterministic model, satisfies

$\text{X(t)=x}{}_0\text{+ ∫}{}^{\mathrm{t}}{}_0\text{∑ lf}{}_1\text{(X(s))ds}$

Under very general conditions lim_N→∞X_N(t)=X(t) a.s. The process X_N(t) is compared to the diffusion processes given by

$\text{Z}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lB}{}_1\text{N∫}{}^{\mathrm{t}}{}_0\text{ft(Z}{}_{\mathrm{N}}\text{(s))ds}$

and

$\text{V(t)=∑ l∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X(s))}\text{d}\text{W}\text{?}{}_1\text{+∫}{}^{\mathrm{t}}{}_0\text{?F(X(s))·V(s)}\text{ds.}$

Under conditions satisfied by most of the applied probability models, it is shown that X_N,Z_N and V can be constructed on the same sample space in such a way that

$\text{X}{}_{\mathrm{N}}\text{(t)=Z}{}_{\mathrm{N}}\text{(t)+O}\text{logN}\text{N}$

and

$\text{N}\text{(X}{}_{\mathrm{N}}\text{(t)?X(t))=V(t)+O}\text{log N}\text{N}$

     

Solution of a system of functional-differential equations arising from an optimization problem

In connection with an optimization problem, all functions ?: Iⁿ → $\text{R}$ with continuous nonzero partial derivatives and satisfying

$\text{??}\text{?x,}{}_{\mathrm{i}}\text{??}\text{?x}{}_{\mathrm{j}}$

for all x_i, x_j ≠ I, i, j = 1,2,…, n (n > 2) are determined (I is an interval of positive real numbers).     

On the number of zeros of diagonal cubic forms

Let M_s, be the number of solutions of the equation

$\text{X}{}_1{}^3\text{+ X}{}_2{}^3\text{+ … + X}{}_{\mathrm{s}}{}^3\text{=0}$

in the finite field GF(p). For a prime p ≡ 1(mod 3),

$\text{∑}\text{s=1}\text{∞}\text{M}{}_{\mathrm{s}}\text{X}{}^{\mathrm{s}}\text{=}\text{x}\text{1 ? px}\text{+}\text{x}{}^2\text{(p ? 1)(2 + dx)}\text{1 ? 3px}{}^2\text{? pdx}{}^3$

,

$\text{M}{}_3\text{= p}{}^2\text{+ d(p ? 1)}$

, and

$\text{M}{}_4\text{= p}{}^2\text{+ 6(p}{}^2\text{? p)}$

. Here d is uniquely determined by

$\text{4}{}_{\mathrm{p}}\text{= d}{}^2\text{+ 27b}{}^2\text{and}\text{d ≡ 1(}\text{mod}\text{3)}$

.     

Analytic inequalities with applications to special functions

Sharp inequalities are derived for certain (polynomial-like) functions of the real variables p_i (i = 1(1)σ) by interpreting p_i as the probabilities that various switches be thrown in certain directions. Parameters m_v in the inequalities are at first taken to be integers; later the inequalities are established when m_v are arbitrary real numbers. The side condition ∑p_i = 1 occurs throughout analysis, so there are many corollaries. Examples of the inequalities established are

$\text{∑}\text{i=1}\text{σ}\text{(1?p}{}_{\mathrm{i}}{}^{\mathrm{m}}\text{)}{}^{\mathrm{m}}\text{>K?1,}$

valid ifm>1

$\text{∑}\text{j=0}\text{r}\text{n}\text{j}\text{p}{}^{\mathrm{jm}}\text{(1?p}{}^{\mathrm{m}}\text{)}{}^{\mathrm{m?j}}\text{+}\text{1?}\text{∑}\text{j=0}\text{r}\text{n}\text{j}\text{p}{}^{\mathrm{j}}\text{(1?p?s)}{}^{\mathrm{n?j}}{}^{\mathrm{m}}\text{> 1+s}{}^{\mathrm{<mtext>max</mtext><mtext>[m,n]</mtext>}}$

valid if m > 1, n > r + 1, 0 < p, s, p + s ? 1, and also valid if $\text{0 < m < 1, 0 < n < r + 1 (1 ? x)}{}^{\mathrm{u}}\text{+ x}{}^{\mathrm{<mtext>1</mtext><mtext>u</mtext>}}\text{< 1,}\text{if}\text{1}\text{2}\text{< x < 1, u > 1}$. (1.03)     

The inertia of a Hermitian matrix having prescribed complementary principal submatrices

For i=1,2 let H_i be a given n_i×n_i Hermitian matrix. We characterize the set of inertias

$\text{In}\text{H}{}_1\text{X}\text{X}{}^1\text{H}{}_2\text{:X}\text{is}\text{n}{}_1\text{×n}{}_2$

in terms of In(H₁) and In(H₂).     

A multivariate correlation ratio

A multivariate correlation ratio of a random vector $\text{Y}$ upon a random vector $\text{X}$ is defined by

$\text{η}{}_{\mathrm{\delta }}\text{(Y;X)={}\text{tr}\text{(δ}{}^{\mathrm{<mtext>?1\; </mtext><mtext>Cov</mtext><mtext>E(Y|X))\}</mtext>}}\text{1}\text{2}\text{{}\text{tr}\text{(δ}{}^{\mathrm{?1}}\text{∑}{}_{\mathrm{Y}}\text{)}}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$

where Λ, a fixed positive definite matrix, is related to the relative importance of predictability for the entries of $\text{Y}$. The properties of η_Λ are discussed, with particular attention paid to a ‘correlation-maximizing’ property. Given are applications of η_Λ to the elliptically symmetric family of distributions and the multinomial distribution. Also discussed is the problem of finding those r linear functions of $\text{Y}$ that are most predictable (in a correlation ratio sense) from $\text{X}$.     

The expected eigenvalue distribution of a large regular graph

Let K₁, K₂,... be a sequence of regular graphs with degree v?2 such that n(X_i)→∞ and ck(X_i)/n(X_i)→0 as i∞ for each k?3, where n(X_i) is the order of X_i, and c_k(X_i) is the number of k- cycles in X₁. We determine the limiting probability density f(x) for the eigenvalues of X>_i as i→∞. It turns out that

$\text{f(x)=}\text{v}\text{4(v?1)?v}{}^2\text{2π(v}{}^2\text{?x}{}^2\text{)}{}_0$

for ?x??2$\text{v-1}$, otherwise It is further shown that f(x) is the expected eigenvalue distribution for every large randomly chosen labeled regular graph with degree v.     

Vector structures and solutions of linear matrix equations

A variety of systems problems give rise to special cases of the linear matrix equation

$\text{∑}\text{i}\text{A}{}_{\mathrm{i}}\text{p×s}\text{X}\text{s×t}\text{B}{}_{\mathrm{i}}\text{t×q}\text{+}\text{∑}\text{j}\text{C}{}_{\mathrm{j}}\text{p×t}\text{X}{}^{\mathrm{T}}\text{t×s}\text{D}{}_{\mathrm{j}}\text{s×q}\text{=}\text{F}\text{p×q}\text{(Z)}$

. F(Z) may be a matrix-valued function of a matrix argument Z. It is the purpose here to summarize and extend some of the applicable solution procedures through a systematic use of operators which convert the matrix equation to vector and dimension-reduced vector forms. The format and details of the results are convenient for machine computation.     

Limit laws for record values

{X_n,n?1} are i.i.d. random variables with continuous d.f. F(x). X_j is a record value of this sequence if X_j>max{X₁,…,X_j?1}. Consider the sequence of such record values {X_Ln,n?1}. Set R(x)=-log(1?F(x)). There exist B_n > 0 such that . in probability (i.p.) iff i.p. iff $\text{{R(kx)?R(x)}}\text{R}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(kx)}$ → ∞ as x→∞ for all k>1. Similar criteria hold for the existence of constants A_n such that X_Ln?A_n → 0 i.p. Limiting record value distributions are of the form N(-log(-logG(x))) where G(·) is an extreme value distribution and N(·) is the standard normal distribution. Domain of attraction criteria for each of the three types of limit laws can be derived by appealing to a duality theorem relating the limiting record value distributions to the extreme value distributions. Repeated use is made of the following lemma: If $\text{P}\text{{X}{}_{\mathrm{n}}\text{?x}=1?e}{}^{\mathrm{-x}}\text{,x?0}$, then X_Ln=Y₀+…+Y_n where the Y_j's are i.i.d. and $\text{P}\text{{Y}{}_{\mathrm{j}}\text{?x}=1?e}{}^{\mathrm{-x}}$.     

Construction of a recurrence relation for modified moments

In this paper we are constructing a recurrence relation of the form

$\text{∑}\text{i=0}\text{r}\text{ω}{}_{\mathrm{i}}\text{(k)m}{}_{\mathrm{k+i}}{}^{\mathrm{\{\lambda \}}}\text{[f] = ω(k)}$

for integrals (called modified moments)

$\text{m}{}_{\mathrm{k}}{}^{\mathrm{\{\lambda \}}}\text{[f]df=}\text{∫}\text{?1}\text{1}\text{f(x)C}{}_{\mathrm{k}}{}^{\mathrm{(\lambda )}}\text{(x)dx (k = 0,1,…)}$

in which C_k^(λ) is the k-th Gegenbauer polynomial of order $\text{λ(λ > ?}\text{1}\text{2}\text{)}$, and f is a function satisfying the differential equation

$\text{∑}\text{i=0}\text{n}\text{P}{}_{\mathrm{i}}\text{(x)f}{}^{\mathrm{(i)}}\text{(x) = p(x) (?1?x?1)}$

of order n, where p₀, p₁, …, p_n ? 0 are polynomials, and m_k^〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense.     

A bijection proving orthogonality of the characters of Sn

Suppose ? and β are partitions of n. If ? ? β, a bijection is given between positive pairs of rim hook tableaux of the same shape λ and content β and ?, respectively, and negative pairs of rim hook tableaux of some other shape μ and content β and ?, respectively. If ? = β, the bijection is between positive pairs and either negative pairs or permutations of hooks. The bijection, in the latter case, is a generalization of the Schensted correspondence between pairs of standard tableaux and permutations. If the irreducible characters of S_n are interpreted combinatorially using the Murnaghan-Nakayama formula, these bijections prove

$\text{∑}\text{λ}\text{X}{}^{\mathrm{\lambda }}{}_{\mathrm{\rho }}\text{X}{}^{\mathrm{\lambda }}{}_{\mathrm{\beta }}\text{=δ}{}_{\mathrm{\rho \beta }}\text{1}{}^{\mathrm{j1}}\text{J1!2}{}^{\mathrm{j2}}\text{J2!…}$

where ? = 1^j12^j2….     

On matrix functions which commute with their derivative

We improve several results published from 1950 up to 1982 on matrix functions commuting with their derivative, and establish two results of general interest. The first one gives a condition for a finite-dimensional vector subspace E(t) of a normed space not to depend on t, when t varies in a normed space. The second one asserts that if A is a matrix function, defined on a set ?, of the form A(t)= U diag(B₁(t),…,B_p(t)) U^-1, t ∈ ?, and if each matrix function B_k has the polynomial form

$\text{B}{}_{\mathrm{k}}\text{(t)=}\text{∑}\text{i=0}\text{α}{}_{\mathrm{k}}\text{f}{}_{\mathrm{ki}}\text{(t)C}{}_{\mathrm{k}}{}^{\mathrm{i}}\text{, t∈ ?, k∈{1,…,p}}$

then A itself has the polynomial form

$\text{A(t)=}\text{∑}\text{i=0}\text{d?1}\text{f}{}_{\mathrm{i}}\text{(t)C}{}^{\mathrm{i}}\text{,t∈?}$

, where

$\text{d=}\text{∑}\text{k=1}\text{p}\text{d}{}_{\mathrm{k}}$

, d_k being the degree of the minimal polynomial of the matrix C_k, for every k ∈ {1,…,p}.     

Representations of differential operators on a Lie group

In this paper we apply the theory of second-order partial differential operators with nonnegative characteristic form to representations of Lie groups. We are concerned with a continuous representation U of a Lie group G in a Banach space $\text{B}$. Let $\text{E}$ be the enveloping algebra of G, and let dU be the infinitesimal homomorphism of $\text{E}$ into operators with the Gårding vectors as a common invariant domain. We study elements in $\text{E}$ of the form

$\text{P=}\text{∑}\text{1}\text{r}\text{X}{}^2{}_{\mathrm{j}}\text{|X}{}_0$

with the X_j,'s in the Lie algebra $\text{G}$.If the elements X₀, X₁,…, X_r generate $\text{G}$ as a Lie algebra then we show that the space of C^∞-vectors for U is precisely equal to the C^∞-vectors for the closure $\text{dU(P)}\text{,}\text{of}\text{dU(P)}$. This result is applied to obtain estimates for differential operators.The operator $\text{dU(P)}$ is the infinitesimal generator of a strongly continuous semigroup of operators in $\text{B}$. If X₀ = 0 we show that this semigroup can be analytically continued to complex time ζ with Re ζ > 0. The generalized heat kernels of these semigroups are computed. A space of rapidly decreasing functions on G is introduced in order to treat the heat kernels.For unitary representations we show essential self-adjointness of all operators $\text{dU(Σ}{}_1{}^{\mathrm{r}}\text{X}{}_{\mathrm{j}}{}^2\text{+ (?1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{X}{}_0$ with X₀ in the real linear span of the X_j's. An application to quantum field theory is given.Finally, the new characterization of the C^∞-vectors is applied to a construction of a counterexample to a conjecture on exponentiation of operator Lie algebras.Our results on semigroups of exponential growth, and on the space of C^∞ vectors for a group representation can be viewed as generalizations of various results due to Nelson-Stinespring [18], and Poulsen [19], who prove essential self-adjointness and a priori estimates, respectively, for the sum of the squares of elements in a basis for $\text{G}$ (the Laplace operator). The work of Hörmander [11] and Bony [3] on degenerate-elliptic (hypoelliptic) operators supplies the technical basis for this generalization. The important feature is that elliptic regularity is too crude a tool for controlling commutators. With the aid of the above-mentioned hypoellipticity results we are able to “control” the (finite dimensional) Lie algebra generated by a given set of differential operators.     

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,I

For (x,y,t)∈$\text{R}$ⁿ × $\text{R}$ⁿ × $\text{R}$, denote $\text{X}{}_{\mathrm{j}}\text{=}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{, y}{}_{\mathrm{j}}\text{=}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}$ and $\text{L}{}_{\mathrm{\alpha }}\text{=?}\text{1}\text{4}\text{∑}\text{j=1}\text{n}\text{X}{}_{\mathrm{j}}{}^2\text{+ Y}{}_{\mathrm{j}}{}^2\text{+}\text{?}\text{?t}$. When α = n ? 2q, $\text{L}$_a represents the action of the Kohn Laplacian □_b on q-forms on the Heisenberg group. For ?n < α < n, we construct a parametrix for the Dirichlet problem in smooth domains D near non-characteristic points of ?D. A point w of ?D is non-characteristic if one of X₁,…, X_n, Y₁,…, Y_n is transverse to ?D at w. This yields sharp local estimates in the Dirichlet problem in the appropriate non-isotropic Lipschitz classes. The main new tool is a “convolution calculus” of pseudo-differential operators that can be applied to the relevant layer potentials, for which the usual asymptotic composition formula is false. Characteristic points are treated in Part II.     

Cardinal interpolation and spline fucntions V. The B-splines for cardinal Hermite interpolation

In the third paper of this series on cardinal spline interpolation [4] Lipow and Schoenberg study the problem of Hermite interpolation

$\text{S(v) = Y}{}_{\mathrm{v}}\text{, S′(v) = Y}{}_{\mathrm{v}}\text{′,…,S}{}^{\mathrm{(r?1)}}\text{(v) = Y}{}_{\mathrm{v}}{}^{\mathrm{(r?1)}}\text{for all}\text{v}$

. The B-splines are there conspicuous by their absence, although they were found very useful for the case γ = 1 of ordinary (or Lagrange) interpolation (see [5–10]). The purpose of the present paper is to investigate the B-splines for the case of Hermite interpolation (γ > 1). In this sense the present paper is a supplement to [4] and is based on its results. This is done in Part I. Part II is devoted to the special case when we want to solve the problem

$\text{S(v) = Y}{}_{\mathrm{v}}\text{, S′(v) = Y}{}_{\mathrm{v}}\text{′}\text{for all}\text{v}$

by quintic spline functions of the class C?(– ∞, ∞). This is the simplest nontrivial example for the general theory. In Part II we derive an explicit solution for the problem (1), where v = 0, 1,…, n.