On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X₁, X₂, …, X_n) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ₁, Θ₂, …, Θ_n). Let θ_x^* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ_x^* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θ_x^* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X₁, X₂, …, X_n is considered for a location vector Θ = (Θ₁, Θ₂, …, Θ_n) as x gets large along a path, γ, in Rⁿ. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞.     

Robust M-estimators of location vectors

Let X₁,…,X_n be i.i.d. random vectors in R^m, let θεR^m be an unknown location parameter, and assume that the restriction of the distribution of X₁−θ to a sphere of radius d belongs to a specified neighborhood of distributions spherically symmetric about 0. Under regularity conditions on and d, the parameter θ in this model is identifiable, and consistent M-estimators of θ (i.e., solutions of Σ_i=1ⁿψ(|X_i− |)(X_i− )=0) are obtained by using “re-descenders,” i.e., ψ's wh satisfy ψ(x)=0 for x≥c. An iterative method for solving for is shown to produce consistent and asymptotically normal estimates of θ under all distributions in . The following asymptotic robustness problem is considered: finding the ψ which is best among the re-descenders according to Huber's minimax variance criterion.     

Central limit theorems in D[0, 1]

Summary Let X be a stochastic process with sample paths in the usual Skorohod space D[0, 1]. For a sequence {X _n} of independent copies of X, let S _n=X₁++X_n. Conditions which are either necessary or sufficient for the weak convergence of n ^–1/2(S _n–ES_n) to a Gaussian process with sample paths in D[0, 1] are discussed. Stochastically continuous processe are considered separately from those with fixed discontinuities. A bridge between the two is made by a Decomposition central limit theorem.     

Best Interpolatory Approximation in Normed Linear Spaces

A theory of best approximation with interpolatory contraints from a finite-dimensional subspaceMof a normed linear spaceXis developed. In particular, to eachxX, best approximations are sought from a subsetM(x) ofMwhichdependson the elementxbeing approximated. It is shown that this “parametric approximation” problem can be essentially reduced to the “usual” one involving a certainfixedsubspaceM₀ofM. More detailed results can be obtained when (1) Xis a Hilbert space, or (2) Mis an “interpolating subspace” ofX(in the sense of [1]).     

Third-order asymptomic properties of a class of test statistics under a local alternative

Suppose that {X_i; I = 1, 2, …,} is a sequence of p-dimensional random vectors forming a stochastic process. Let p_n, θ(X_n), X_n ^np, be the probability density function of X_n = (X₁, …, X_n) depending on θ Θ, where Θ is an open set of ¹. We consider to test a simple hypothesis H : θ = θ₀ against the alternative A : θ ≠ θ₀. For this testing problem we introduce a class of tests , which contains the likelihood ratio, Wald, modified Wald, and Rao tests as special cases. Then we derive the third-order asymptotic expansion of the distribution of T under a sequence of local alternatives. Using this result we elucidate various third-order asymptotic properties of T (e.g., Bartlett's adjustments, third-order asymptotically most powerful properties). Our results are very general, and can be applied to the i.i.d. case, multivariate analysis, and time series analysis. Two concrete examples will be given. One is a Gaussian ARMA process (dependent case), and the other is a nonlinear regression model (non-identically distributed case).     

Solution of the Ulam Stability Problem for Euler–Lagrange Quadratic Mappings

In 1940 S. M. Ulam proposed at the University of Wisconsin theproblem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist.” In 1968 S. U. Ulam proposed the moregeneral problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” In 1978 P. M. Gruber proposed theUlam type problem: “Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this object by objects, satisfying the property exactly?” According to P. M. Gruber this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1982–1996 we solved the above Ulam problem, or equivalently the Ulam type problem for linear mappings and established analogous stability problems. In this paper we first introduce newquadratic weighted meansandfundamental functional equationsand then solve theUlam stability problemfornon-linear Euler–Lagrange quadratic mappingsQ:X → Y, satisfying a mean equation and functional equation[formula]for all 2-dimensional vectors (x₁, x₂) X², withXa normed linear space (Y a real complete normed linear space), and any fixed pair (a₁, a₂) of realsa_iand any fixed pair (m₁, m₂) of positive realsm_i(i = 1, 2), [formula]     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

Effacement des dérivations et spectres premiers des algèbres quantiques

Given any (commutative) field k and any iterated Ore extension R=k[X₁][X₂;σ₂,δ₂][X_N;σ_N,δ_N] satisfying some suitable assumptions, we construct the so-called “Derivative-Elimination Algorithm.” It consists of a sequence of changes of variables inside the division ring F=Fract(R), starting with the indeterminates (X₁,…,X_N) and terminating with new variables (T₁,…,T_N). These new variables generate some quantum-affine space such that . This algorithm induces a natural embedding which satisfies the following property:

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. We study both the derivative-elimination algorithm and natural embedding and use them to produce, for the general case, a (common) proof of the “quantum Gelfand–Kirillov” property for the prime homomorphic images of the following quantum algebras: , (wW), R_q[G] (where G denotes any complex, semi-simple, connected, simply connected Lie group with associated Lie algebra and Weyl group W), quantum matrices algebras, quantum Weyl algebras and quantum Euclidean (respectively symplectic) spaces. Another application will be given in [G. Cauchon, J. Algebra, to appear]: In the general case, the prime spectrum of any quantum matrices algebra satisfies the normal separation property.     

On the Maximal Multiplicity of Parts in a Random Integer Partition

We study the asymptotic behavior of the maximal multiplicity μ_n = μ_n(λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμ_n/(6n)^1/2 converges weakly to max _jX_j/j as n→∞, where X₁, X₂, … are independent and exponentially distributed random variables with common mean equal to 1.2000 Mathematics Subject Classification: Primary—05A17; Secondary—11P82, 60C05, 60F05     

On the Rate of Decay of the Concentration Function of the Sum of Independent Random Variables

Let X₁, ... , X_n be i.i.d. integral valued random variables and S_n their sum. In the case when X₁ has a moderately large tail of distribution, Deshouillers, Freiman and Yudin gave a uniform upper bound for max _{k ∊ ℤ} Pr{S_n = k} (which can be expressed in term of the Lévy Doeblin concentration of S_n), under the extra condition that X₁ is not essentially supported by an arithmetic progression. The first aim of the paper is to show that this extra condition cannot be simply ruled out. Secondly, it is shown that if X₁ has a very large tail (larger than a Cauchy-type distribution), then the extra arithmetic condition is not sufficient to guarantee a uniform upper bound for the decay of the concentration of the sum S_n. Proofs are constructive and enhance the connection between additive number theory and probability theory.À Jean-Louis Nicolas, avec amitié et respect2000 Mathematics Subject Classification: Primary—60Fxx, 60Exx, 11Pxx, 11B25     

Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

For X one observation on a p-dimensional (p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L(δ, θ) = (δ − θ)′ D(δ − θ) where D is a known p × p positive definite matrix. For C a p × p known positive definite matrix, conditions are given under which estimators of the form δ_a,r,C,D(X) = (I − (ar(|X|²)) D^−1/2CD^1/2 |X|⁻²)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X₁, X₂, …, X_n are taken on a p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(X₁, X₂, …, X_n), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.     

A self-normalized Erd s—Rényi type strong law of large numbers

The original Erd s—Rényi theorem states that max_0kn∑^k+[clogn]_i=k+1X_i/[clogn]→α(c),c>0, almost surely for i.i.d. random variables {X_n, n1} with mean zero and finite moment generating function in a neighbourhood of zero. The latter condition is also necessary for the Erd s—Rényi theorem, and the function α(c) uniquely determines the distribution function of X₁. We prove that if the normalizing constant [c log n] is replaced by the random variable ∑^k+[clogn]_i=k+1(X²_i+1), then a corresponding result remains true under assuming only the exist first moment, or that the underlying distribution is symmetric.     

Subtractive Categories

We introduce a notion of a subtractive category. It generalizes the notion of a pointed subtractive variety of universal algebras in the sense of A. Ursini. Subtractive categories are closely related to Mal’tsev and additive categories: (i) a category C with finite limits is a Mal’tsev category if and only if for every object X in C the category Pt(X)=((X,1_X)↓(C↓X)) of “points over X” is subtractive; (ii) a pointed category C with finite limits is additive if and only if C is subtractive and half-additive.Mathematics Subject Classifications (2000) 18C99, 18E05, 08B05.     

L1-optimal estimates for a regression type function in R

Let X₁, X₂, …, X_n be random vectors that take values in a compact set in R^d, d ≥ 1. Let Y₁, Y₂, …, Y_n be random variables (“the responses”) which conditionally on X₁ = x₁, …, X_n = x_n are independent with densities f(y | x_i, θ(x_i)), i = 1, …, n. Assuming that θ lives in a sup-norm compact space Θ_q,d of real valued functions, an optimal L₁-consistent estimator of θ is constructed via empirical measures. The rate of convergence of the estimator to the true parameter θ depends on Kolmogorov's entropy of Θ_q,d.     

Reverse mathematics and well-ordering principles: A pilot study

The larger project broached here is to look at the generally sentence “if X is well-ordered then f(X) is well-ordered”, where f is a standard proof-theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded ω-models for a particular theory T_f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, we prove in this paper that the statement “if X is well-ordered then ε_X is well-ordered” is equivalent to . This was first proved by Marcone and Montalban [Alberto Marcone, Antonio Montalbán, The epsilon function for computability theorists, draft, 2007] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schütte’s method of proof search (deduction chains) [Kurt Schütte, Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977] and cut elimination for a (small) fragment of .     

Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform

An n-dimensional random vector X is said (Cambanis, S., Keener, R., and Simons, G. (1983). J. Multivar. Anal., 13 213–233) to have an α-symmetric distribution, α > 0, if its characteristic function is of the form φ(|ξ₁|^α + … + |ξ_n|^α). Using the Radon transform, integral representations are obtained for the density functions of certain absolutely continuous α-symmetric distributions. Series expansions are obtained for a class of apparently new special functions which are encountered during this study. The Radon transform is also applied to obtain the densities of certain radially symmetric stable distributions on ⁿ. A new class of “zonally” symmetric stable laws on ⁿ is defined, and series expansions are derived for their characteristic functions and densities.     

Stochastic integration on partially ordered sets

The stochastic integral is introduced with respect to a stochastic process X = (X_s)_sεV, where V is any general partially ordered set satisfying some mild regularity conditions. As important examples the stochastic integral is constructed with respect to a class of Gaussian processes having similarities to the Brownian motion on the real line, and also with respect to L²-martingales under an assumption of conditional independence on the underlying σ-fields.     

The Injective Spectrum of a Noncommutative Space

For a noncommutative space X, we study Inj(X), the set of isomorphism classes of indecomposable injective X-modules. In particular, we look at how this set, suitably topologized, can be viewed as an underlying “spectrum” for X. As applications we discuss noncommutative notions of irreducibility and integrality, and a way of associating an integral subspace of X to each element of Inj(X) which behaves like a “weak point.”     

Set-Indexed Conditional Empirical and Quantile Processes Based on Dependent Data

We consider a conditional empirical distribution of the form F_n(C x)=∑ⁿ_t=1 ω_n(X_t−x) I{Y_tC} indexed by C , where {(X_t, Y_t), t=1, …, n} are observations from a strictly stationary and strong mixing stochastic process, {ω_n(X_t−x)} are kernel weights, and is a class of sets. Under the assumption on the richness of the index class in terms of metric entropy with bracketing, we have established uniform convergence and asymptotic normality for F_n(· x). The key result specifies rates of convergences for the modulus of continuity of the conditional empirical process. The results are then applied to derive Bahadur–Kiefer type approximations for a generalized conditional quantile process which, in the case with independent observations, generalizes and improves earlier results. Potential applications in the areas of estimating level sets and testing for unimodality (or multimodality) of conditional distributions are discussed.     

On the uniform distribution of sequences of random variables

Summary This paper deals with the almost sure uniform distribution (modulo 1) of sequences of random variables. In the case where the law of the increments X _n+h –X _n of the sequence X ₀, X ₁, does not depend on n, sufficient conditions are given to assure the uniform distribution (modulo 1) with probability one. As an illustrative example the partial sums of a sequence of independent, identically distributed variables is considered.