Limit Laws for Norms of IID Samples with Weibull Tails

We are concerned with the limit distribution of l _t -norms (of order t) of samples of i.i.d. positive random variables, as N→∞, t→∞. The problem was first considered by Schlather [(2001), Ann. Probab. 29, 862–881], but the case where {X _i } belong to the domain of attraction of Gumbel’s double exponential law (in the sense of extreme value theory) has largely remained open (even for an exponential distribution). In this paper, it is assumed that the log-tail distribution function is regularly varying at infinity with index . We proceed from studying the limit distribution of the sums , which is of interest in its own right. A proper growth scale of N relative to t appears to be of the form (). We show that there are two critical points, α₁ = 1 and α₂ = 2, below which the law of large numbers and the central limit theorem, respectively, break down. For α < 2, under a slightly stronger condition of normalized regular variation of h, we prove that the limit laws for S _N (t) are stable, with characteristic exponent and skewness parameter . A complete picture of the limit laws for the norms R _N (t) = S _N (t)^1/t is then derived. In particular, our results corroborate a conjecture in Schlather [(2001), Ann. Probab. 29, 862–881] regarding the “endpoints” , α→ 0.      

Uniform Comparison of Tails of (Non-Symmetric) Probability Measures and Their Symmetrized Counterparts with Applications

Let (, ) be a separable Banach space and let be a class of probability measures on , and let denote the symmetrization of . We provide two sufficient conditions (one in terms of certain quantiles and the other in terms of certain moments of relative to μ and , ) for the “uniform comparison” of the μ and measure of the complements of the closed balls of centered at zero, for every . As a corollary to these “tail comparison inequalities,” we show that three classical results (the Lévy-type Inequalities, the Kwapień-Contraction Inequality, and a part of the It?–Nisio Theorem) that are valid for the symmetric (but not for the general non-symmetric) independent -valued random vectors do indeed hold for the independent random vectors whose laws belong to any which satisfies one of the two noted conditions and which is closed under convolution. We further point out that these three results (respectively, the tail comparison inequalities) are valid for the centered log-concave, as well as, for the strictly α-stable (or the more general strictly (r, α) -semistable) α ≠ 1 random vectors (respectively, probability measures). We also present several examples which we believe form a valuable part of the paper.      

The Monge Problem in Banach Spaces

In this paper, we generalize the Kantorovich functional to K?the-spaces for a cost or a profit function. We examine the convergence of probabilities with respect to this functional for some K?the-spaces. We study the Monge problem: Let be a K?the-space, P and Q two Borel probabilities defined on a Polish space M and a cost function . A K?the functional is defined by (P, Q) = inf where is the law of X. If c is a profit function, we note . (P, Q) = sup Under some conditions, we show the existence of a Monge function, φ, such that , or .      

Poincaré inequality and Palais–Smale condition for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

An improved Poincaré inequality and validity of the Palais-Smale condition are investigated for the energy functional on , 1 < p < ∞, where Ω is a bounded domain in , is a spectral (control) parameter, and is a given function, in Ω. Analysis is focused on the case λ = λ₁, where −λ₁ is the first eigenvalue of the Dirichlet p-Laplacian Δ _p on , λ₁ > 0, and on the “quadratization” of within an arbitrarily small cone in around the axis spanned by , where stands for the first eigenfunction of Δ _p associated with −λ₁.     

Pointwise estimates for relative fundamental solutions of heat equations in $${\mathbb{R} \times \mathbb{C}}$$

Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ^∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.      

On The Invariant Measure of a Positive Recurrent Diffusion in $${\mathbb{R}}$$

Given a one-dimensional positive recurrent diffusion governed by the Stratonovich SDE , we show that the associated stochastic flow of diffeomorphisms focuses as fast as , where is the finite stationary measure. Moreover, if the drift is reversed and the diffeomorphism is inverted, then the path function so produced tends, independently of its starting point, to a single (random) point whose distribution is . Applications to stationary solutions of X _t , asymptotic behavior of solutions of SPDEs and random attractors are offered. This paper was written while the author was visiting Northwestern University and the opinions expressed in it are those of the author alone and do not necessarily reflect the views of Merrill Lynch, its subsidiaries or affiliates.     

Normal subgroups of nonstandard symmetric and alternating groups

Let be a nonstandard model of Peano Arithmetic with domain M and let be nonstandard. We study the symmetric and alternating groups S _n and A _n of permutations of the set internal to , and classify all their normal subgroups, identifying many externally defined such normal subgroups in the process. We provide evidence that A _n and S _n are not split extensions by these normal subgroups, by showing that any such complement if it exists, cannot be a limit of definable sets. We conclude by identifying an -valued metric on and (where B _S , B _A are the maximal normal subgroups of S _n and A _n identified earlier) making these groups into topological groups, and by showing that if is -saturated then and are complete with respect to this metric.      

A Notion of Stopping Line for Set-Indexed Processes

In this paper, we introduce the class of -stopping lines which generalize the planar stopping lines in Merzbach [(1980), Stochastic Process. Appl. 10, 49–63] by replacing the positive quadrant of the plane by a collection of compact subsets of a fixed topological space. Our notion of stopping line also compliments and generalizes the stopping sets defined in Ivanoff and Merzbach [(1995), Stochastic Process. Appl. 57, 83–98].      

A note on curvature of α-connections of a statistical manifold

The family of α-connections ∇^(α) on a statistical manifold equipped with a pair of conjugate connections and is given as . Here, we develop an expression of curvature R ^(α) for ∇^(α) in relation to those for . Immediately evident from it is that ∇^(α) is equiaffine for any when are dually flat, as previously observed in Takeuchi and Amari (IEEE Transactions on Information Theory 51:1011–1023, 2005). Other related formulae are also developed. The work was conducted when the author was on sabbatical leave as a visiting research scientist at the Mathematical Neuroscience Unit, RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan.     

Euler homology

We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented cobordism ring of a topological space X. This homology theory Eh _* has coefficients in every nonnegative dimension. There exists a natural transformation that for X = pt assigns to each smooth manifold its Euler characteristic mod 2. The homology theory is constructed using cobordism of stratifolds, which are singular objects defined below. An isomorphism of graded -modules is shown for any CW-complex X. For discrete groups G, we also define an equivariant version of the homology theory Eh _*, generalizing the equivariant Euler characteristic.     

Unstable extremal surfaces of the “Shiffman functional” spanning rectifiable boundary curves

In this paper we derive a sufficient condition for the existence of extremal surfaces of a parametric functional with a dominant area term, which do not furnish global minima of within the class of H ^1,2-surfaces spanning an arbitrary closed rectifiable Jordan curve that merely has to satisfy a chord-arc condition. The proof is based on the “mountain pass result” of (Jakob in Calc Var 21:401–427, 2004) which yields an unstable -extremal surface bounded by an arbitrary simple closed polygon and Heinz’ ”approximation method” in (Arch Rat Mech Anal 38:257–267, 1970). Hence, we give a precise proof of a partial result of the mountain pass theorem claimed by Shiffman in (Ann Math 45:543–576, 1944) who only outlined a very sketchy and partially incorrect proof.     

Skorohod representation on a given probability space

Let $(\Omega,\mathcal{A},P)Let be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and an arbitrary map, n = 1,2,.... If μ is tight and X _n converges in distribution to μ (in Hoffmann–J?rgensen’s sense), then X∼μ for some S-valued random variable X on . If, in addition, the X _n are measurable and tight, there are S-valued random variables and X, defined on , such that , X∼μ, and a.s. for some subsequence (n _k ). Further, a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X _n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to for some H⊂Ω with P ^*(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken , for some H⊂ (0,1) with outer Lebesgue measure 1, where is the Borel σ-field on (0,1) and m _H the only extension of Lebesgue measure such that m _H (H) = 1. In order to prove the previous results, it is also shown that, if X _n converges in distribution to a separable limit, then X _{n k} converges stably for some subsequence (n _k ).      

Sums of random symmetric matrices and quadratic optimization under orthogonality constraints

Let B _i be deterministic real symmetric m × m matrices, and ξ _i be independent random scalars with zero mean and “of order of one” (e.g., ). We are interested to know under what conditions “typical norm” of the random matrix is of order of 1. An evident necessary condition is , which, essentially, translates to ; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S _N is : for all Ω > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints , where F is quadratic in X = (X ₁,... ,X _k ). We show that when F is convex in every one of X _j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices . Research was partly supported by the Binational Science Foundation grant #2002038.     

Transition Semi–groups on a Local Field Induced by Galois Group and their Representation

For a given map defined on the field of p-adic numbers satisfying

On a singular perturbation problem related to optimal lifting in BV-space

We prove a Γ-convergence result for the family of functionals defined on H ¹(Ω) by for a given and a parameter . We show that in either of the two cases, p = 2 or , any limit of the minimizers is an optimal lifting.     

Refinement of Convergence Rates for Tail Probabilities

Let X₁, X₂,... be, i.i.d. random variables, and put . We find necessary and sufficient moment conditions for , where δ≥ 0 and q>0, and with a_n>0 and b_n is either or The series f(x) we deal with are classical series studied by Hsu and Robbins, Erdős, Spitzer, Baum and Katz, Davis, Lai, Gut, etc     

Inertial Iterative Process for Fixed Points of Certain Quasi-nonexpansive Mappings

This paper deals with a general formalism which consists in approximating a point in a nonempty set , in a real Hilbert space , by a sequence such that , where , are in and is a sequence included in a certain class of self-mappings on , such that every fixed point set of contains . This iteration method is inspired by an implicit discretization of the second order ‘heavy ball with friction’ dynamical system. Under suitable conditions on the parameters and the operators , we prove that this scheme generates a sequence which converges weakly to an element of . In particular, by appropriate choices of , this algorithm works for approximating common fixed points of infinite countable families of a wide class of operators which includes -averaged quasi-nonexpansive mappings for .      

Base subsets of symplectic Grassmannians

Let V and V′ be 2n-dimensional vector spaces over fields F and F′. Let also Ω: V× V→ F and Ω′: V′× V′→ F′ be non-degenerate symplectic forms. Denote by Π and Π′ the associated (2n−1)-dimensional projective spaces. The sets of k-dimensional totally isotropic subspaces of Π and Π′ will be denoted by and ${\mathcal G}'_{k}$, respectively. Apartments of the associated buildings intersect and by so-called base subsets. We show that every mapping of to sending base subsets to base subsets is induced by a symplectic embedding of Π to Π′.     

Weak convergence of diffusion processes on Wiener space

Let γ be a Gaussian measure on a Suslin space X, H be the corresponding Cameron–Martin space and {e _i } ⊂ H be an orthonormal basis of H. Suppose that μ _n = ρ _n · γ is a sequence of probability measures which converges weakly to a probability measure μ = ρ · γ Consider a sequence of Dirichlet forms , where and . We prove some sufficient conditions for Mosco convergence where . In particular, if X is a Hilbert space, and can be uniformly approximated by finite dimensional conditional expectations for every fixed e _i , then under broad assumptions Mosco and the distributions of the associated stochastic processes converge weakly.