α-Stable characterization of Banach spaces (1 < α < 2)

This work characterizes some subclasses of α-stable (0 < α < 1) Banach spaces in terms of the extendibility to Radon laws of certain α-stable cylinder measures. These result extend the work of S. Chobanian and V. Tarieladze (J. Multivar. Anal.7, 183–203 (1977)). For these spaces it is shown that every Radon stable measure is the continuous image of a stable measure on a suitable L_β space with β = α(1 − α)⁻¹. The latter result extends some work of Garling (Ann. Probab.4, 600–611 (1976)) and Jain (Proceedings, Symposia in Pure Math. XXXI, p. 55–65, Amer. Math. Soc., Providence, R.I.).     

Log-fractional stable processes

The first problem attacked in this paper is answering the question whether all 1/-self-similar -stable processes with stationary increments are -stable motions. The answer is yes for = 2, no for 1<2 and unknown for 0<<1. We single out the log-fractional stable processes for 1<2, different from -stable motions for ≠2. They can be regarded as the limit of fractional stable processes as the exponent in the kernel tends to 0. The paper ends with a limit theorem for partial sum processes of moving averages of iid random variables in the domain of attraction of a strictly stable law, with log-fractional stable processes as limits in law. The conditions involve de Haan's class Π of slowly varying functions.     

Strictly operator-stable distributions

Strictly operator-stable distributions are defined and discussed. Characterization of strictly stable distributions with exponent 1 is generalized to strictly (α, Q)-stable distributions with α being an eigenvalue of Q.     

α-Stable limit theorems for sums of dependent random vectors

Several α-stable limit theorems for sums of dependent random vectors are proved via point processes theory; p-mixing, m-dependence, and the type of mixing treated within the extreme value theory are considered.     

Stable limits of empirical processes of moving averages with infinite variance

The paper obtains a functional limit theorem for the empirical process of a stationary moving average process X_t with i.i.d. innovations belonging to the domain of attraction of a symmetric -stable law, 1<<2, with weights b_j decaying as j^−β, 1<β<2/. We show that the empirical process (normalized by N^1/β) weakly converges, as the sample size N increases, to the process c_x⁺L⁺+c_x⁻L⁻, where L⁺,L⁻ are independent totally skewed β-stable random variables, and c_x⁺,c_x⁻ are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(X_s) is an β-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/<β<1 and 2/<β studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.     

On αrγs(k)-perfect graphs

For some integer k0 and two graph parameters π and τ, a graph G is called πτ(k)-perfect, if π(H)−τ(H)k for every induced subgraph H of G. For r1 let α_r and γ_r denote the r-(distance)-independence and r-(distance)-domination number, respectively. In (J. Graph Theory 32 (1999) 303–310), I. Zverovich gave an ingenious complete characterization of α₁γ₁(k)-perfect graphs in terms of forbidden induced subgraphs. In this paper we study α_rγ_s(k)-perfect graphs for r,s1. We prove several properties of minimal α_rγ_s(k)-imperfect graphs. Generalizing Zverovich's main result in (J. Graph Theory 32 (1999) 303–310), we completely characterize α_2r−1γ_r(k)-perfect graphs for r1. Furthermore, we characterize claw-free α₂γ₂(k)-perfect graphs.     

Note on multidimensional empirical processes for φ-mixing random vectors

In 1974, Sen proved weak convergence of the empirical processes (in the J₁-topology on D^p[0, 1]) for a stationary φ-mixing sequence of stochastic p( 1)-vectors. In this note, we show that Sen's theorem on weak convergence of the multidimensional empirical process for a stationary φ-mixing sequence of stochastic vectors remains true under a less restrictive condition on the mixing constants {φ_n}, i.e., φ_n = O(n^−1−δ) for some δ > 0.     

Test of association between multivariate stable vectors

Based on observations of d-dimensional random vectors in the domain of attraction of a stable distribution with (multi-)index α = (α₁, …, α_d), an estimator for the dependence function of the α_i-stable variables is constructed. The estimator utilizes the α-tail-estimator and an estimator of the spectral measure of the α-stable law. This estimator gives rise to a test of association of the stable components and various quantitative measures of association.     

Graphs satisfying inequality θ(G)θ(G)

In this paper, we study the edge clique cover number of squares of graphs. More specifically, we study the inequality θ(G²)θ(G) where θ(G) is the edge clique cover number of a graph G. We show that any graph G with at most θ(G) vertices satisfies the inequality. Among the graphs with more than θ(G) vertices, we find some graphs violating the inequality and show that dually chordal graphs and power-chordal graphs satisfy the inequality. Especially, we give an exact formula computing θ(T²) for a tree T.     

On stochastic integral representation of stable processes with sample paths in Banach spaces

Certain path properties of a symmetric α-stable process X(t) = ∫_Sh(t, s) dM(s), t T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double α-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624–632) is extensively used and the relationship between these two representations is discussed.     

Random-Time Isotropic Fractional Stable Fields

Generalizing both Substable Fractional Stable Motions (FSMs) and Indicator FSMs, we introduce α-stabilized subordination, a procedure which produces new FSMs (H-self-similar, stationary increment symmetric α-stable processes) from old ones. We extend these processes to isotropic stable fields which have stationary increments in the strong sense, i.e., processes which are invariant under Euclidean rigid motions of the multi-dimensional time parameter. We also prove a Stable Central Limit Theorem which provides an intuitive picture of α-stabilized subordination. Finally we show that α-stabilized subordination of Linear FSMs produces null-conservative FSMs, a class of FSMs introduced by Samorodnitsky (Ann. Probab. 33(5):1782–1803, 2005).     

Weak Limits for Multivariate Random Sums

Let {Xi, i1} be a sequence of i.i.d. random vectors inRd, and letνp, 0<p<1, be a positive, integer valued random variable, independent ofXis. Theν-stable distributions are the weak limits of properly normalized random sums ∑νpi=1 Xiasνp ∞ andpνp ν. We study the properties ofν-stable laws through their representation via stable laws. In particular, we estimate their tail probabilities and provide conditions for finiteness of their moments.     

Functions of class Lip(α, p) and their Taylor mean

The object of this paper is to study the rapidity of convergence of the Taylor mean of the Fourier series of ƒ(x) when ƒ(x) belongs to the class Lip(α, p). We show that it is of Jackson order provided that a suitable integrability condition is imposed upon the function

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Almost Kenmotsu manifolds with a condition of η-parallelism

We consider almost Kenmotsu manifolds (M²ⁿ⁺¹,φ,ξ,η,g) with η-parallel tensor h^′=h○φ, 2h being the Lie derivative of the structure tensor φ with respect to the Reeb vector field ξ. We describe the Riemannian geometry of an integral submanifold of the distribution orthogonal to ξ, characterizing the CR-integrability of the structure. Under the additional condition _ξh^′=0, the almost Kenmotsu manifold is locally a warped product. Finally, some lightlike structures on M²ⁿ⁺¹ are introduced and studied.     

On the distributions of the form ∑i (δpi−δni)

We present some properties of the distributions T of the form ∑_i (δ_pi−δ_ni), with ∑_i d(p_i,n_i)<∞, which arise in the study of the 3-d Ginzburg–Landau problem; see Bourgain et al. (C. R. Acad. Sci. Paris, Ser. I 331 (2000) 119–124). We show that there always exists an irreducible representation of T. We also extend a result of Smets (C. R. Acad. Sci. Paris, Ser. I 334 (2002) 371–374) which says that T is a measure iff T can be written as a finite sum of dipoles.     

Polynomial Projections inC[−1, 1] andL(−1, 1) with Growthn, 0<γ1/2

Forγ(0, 1/2] we constructn-dimensional polynomial subspacesY_nofC[−1, 1] andL¹(−1, 1) such that the relative projection constantsλ(Y_n, C[−1, 1]) andλ(Y_n, L¹(−1, 1)) grow asn^γ. These subspaces are spanned by Chebyshev polynomials of the first and second kind, respectively. The spacesL¹_{w(α, β}wherew_{α, β}is the weight function of the Jacobi polynomials and (α, β){(−1/2, −1/2), (−1/2, 0), (0, −1/2)} are also studied.     

A note on a theorem of Berkes and Philipp for dependent sequences

We improve an almost sure invariance principle for f-mixing sequences of real random variables with finite (2 + δ)th moment (0 < δ 2) due to Berkes and Philipp (1979). The exponent of the slow logarithmic rate of mixing in that theorem is relaxed from 160/δ to (1 + ε)(1 + 2/δ) and the undetermined quantities a_N are replaced by Nσ² for some σ > 0.     

The heat flow in nonlinear Hodge theory

We study the nonlinear Hodge system dω=0 and δ(ρ(|ω|²)ω)=0 for an exterior form ω on a compact oriented Riemannian manifold M, where ρ(Q) is a given positive function. The solutions are called ρ-harmonic forms. They are the stationary points on cohomology classes of the functional with e′(Q)=ρ(Q)/2. The ρ-codifferential of a form ω is defined as δ_ρω=ρ⁻¹δ(ρω) with ρ=ρ(|ω|²).We evolve a given closed form ω₀ by the nonlinear heat flow system for a time-dependent exterior form ω(x,t) on M. This system is the differential of the normalized gradient flow for E(ω) with ω=ω₀+du. Under a technical assumption on the function 2ρ′(Q)Q/ρ(Q), we show that the nonlinear heat flow system , with initial condition ω(·,0)=ω₀, has a unique solution for all times, which converges to a ρ-harmonic form in the cohomology class of ω₀. This yields a nonlinear Hodge theorem that every cohomology class of M has a unique ρ-harmonic representative.     

Property (quasi‐α) and the denseness of norm attaining mappings

We introduce property (quasi ‐α), which implies property (A) defined by Lindenstrauss [10] and whose dual property is property (quasi‐β) [2]. We consider relations between this property and other sufficient conditions for property (A), and study the denseness of norm attaining mappings under the conditions of these properties. In particular, if each of the Banach spaces X_k , 1 ≤ k ≤ n – 1, has property (quasi‐α) and X_n has property (A), then the projective tensor product X₁ ··· X_n has property (A). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)