Mixing normal approximations of vectors of sums and maximum sums

Summary The conditioned central limit theorem for the vector of maximum partial sums based on independent identically distributed random vectors is investigated and the rate of convergence is discussed. The conditioning is that of Rényi (1958,Acta Math. Acad. Sci. Hungar.,9, 215–228). Analogous results for the vector of partial sums are obtained. University of Petroleum and Minerals     

Inequalities of Maximum of Partial Sums and Weak Convergence for a Class of Weak Dependent Random Variables

In this paper, we establish a Rosenthal-type inequality of the maximum of partial sums for ρ^- -mixing random fields. As its applications we get the Hájeck -Rènyi inequality and weak convergence of sums of ρ^- -mixing sequence. These results extend related results for NA sequence and p^＊ -mixing random fields,     

Almost all graphs are rigid - Revisited

A graph is called asymmetric if it has the identity mapping as its only automorphism. In [P. Erdõs, A. Rényi, Asymmetric Graphs, Acta Math. Acad. Sci. Hungar. 14 (1963) 295–315], P. Erdõs and A. Rényi have proven that almost all graphs are asymmetric. A graph is called rigid if it has the identity mapping as its only endomorphism, which is a stronger property than asymmetry. By adopting the approach of Erdõs and Rényi, it is shown that almost all graphs are rigid. A different proof of that result has already been published in [V. Koubek, V. Rödl, On the Minimum Order of Graphs with Given Semigroup, J. Combin. Theory Ser. B 36 (1984) 135–155] (as well as in [P. Hell, J. Nešetřil, Graphs and Homomorphisms, Oxford U. Press, Oxford, 2004]).     

Torsion Completeness of Sylow <Emphasis Type="Italic">p</Emphasis>-Groups in Semisimple Group Rings

Let G be an abelian p-group and K be a field of the first kind with respect to p of char K ≠p and of sp(K) = N or NU {0}. Then it is shown that the normed Sylow p-subgroup S(KG) is torsion complete if and only if G is bounded (Theorem 1). An analogous fact is proved for the case when K is of the second kind (Theorem 2). These completely settle a conjecture posed by us in Compt. Rend. Acad. Bulg. Sci. (1993) and are also a supplement to our result in the modular case published in Acta Math. Hungar. (1997).     

Characteristic properties of large subgroups in primary abelian groups

SupposeG is an arbitrary additively written primary abelian group with a fixed large subgroupL. It is shown thatG is (a) summable; (b) σ-summable; (c) a Σ-group; (d) p^ω+1-projecrive only when so isL. These claims extend results of such a kind obtained by Benabdallah, Eisenstadt, Irwin and Poluianov,Acta Math. Acad. Sci. Hungaricae (1970) and Khan,Proc. Indian Acad. Sci. Sect. A (1978).     

Some new limit theorems for vector space- and group-valued random variables

It is shown that weakly convergent sums (products) of normalized i.i.d. random variables with values in a finite-dimensional vector space or in a group are mixing in the sense of A. Rényi, and limit theorems for random sums with nonindependent indices are obtained. A new version of H. Robbins' limit theorem for random sums is presented. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló Hungary, 1997, Part III.     

Generalized entropy of the Heisenberg spin chain

We consider the XY quantum spin chain in a transverse magnetic field. We consider the Rényi entropy of a block of neighboring spins at zero temperature on an infinite lattice. The Rényi entropy is essentially the trace of some power α of the density matrix of the block. We calculate the entropy of the large block in terms of Klein’s elliptic λ-function. We study the limit entropy as a function of its parameter α. We show that the Rényi entropy is essentially an automorphic function with respect to a certain subgroup of the modular group. Using this, we derive the transformation properties of the Rényi entropy under the map α → α ⁻¹ .     

Rényi entropies of aperiodic dynamical systems

In this paper we continue the study of Rényi entropies of measure-preserving transformations started in [22]. We have established there that for ergodic transformations with positive entropy, the Rényi entropies of orderq, q ∈ ℝ, are equal to either plus infinity (q < 1), or to the measure-theoretic (Kolmogorov-Sinai) entropy (q ≥ 1). The answer for non-ergodic transformations is different: the Rényi entropies of orderq > 1 are equal to the essential infimum of the measure-theoretic entropies of measures forming the decomposition into ergodic components. Thus, it is possible that the Rényi entropies of orderq > 1 are strictly smaller than the measure-theoretic entropy, which is the average value of entropies of ergodic components. This result is a bit surprising: the Rényi entropies are metric invariants, which are sensitive to ergodicity. The proof of the described result is based on the construction of partitions with independent iterates. However, these partitions are obtained in different ways depending onq: forq > 1 we use a version of the well-known Sinai theorem on Bernoulli factors for the non-ergodic transformations; forq < 1 we use the notion of collections of independent sets in Rokhlin-Halmos towers and their properties.     

Lambda composition

A lambda in a graph G is two edges uv and vw such that uw is not an edge. A subgraph A of G is called a lambda-subgraph if every lambda of G has both or neither of its edges in A. We describe the decomposition of a graph into its lambda subgraphs and use this to prove a decomposition theorem of Gallai (Acta Math. Acad. Sci. Hungar. 18 (1967), 25–66). A corollary is that a graph is perfect if and only if each of its edge-minimal lambda subgraphs is. © 1997 John Wiley & Sons, Inc. J Graph Theory 26:9–16, 1997     

The Bohman‐Frieze process near criticality

The Erd?s‐Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erd?s and Rényi states that the Erd?s‐Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erd?s and Rényi, various random graph models have been introduced and studied. In this paper we study the Bohman‐Frieze process, a simple modification of the Erd?s‐Rényi process. The Bohman‐Frieze process also begins with an empty graph on n vertices. At each step two random edges are presented, and if the first edge would join two isolated vertices, it is added to a graph; otherwise the second edge is added. We present several new results on the phase transition of the Bohman‐Frieze process. We show that it has a qualitatively similar phase transition to the Erd?s‐Rényi process in terms of the size and structure of the components near the critical point. We prove that all components at time t_c ? ? (that is, when the number of edges are (t_c ? ?)n/2) are trees or unicyclic components and that the largest component is of size Ω(?^‐2log n). Further, at t_c + ?, all components apart from the giant component are trees or unicyclic and the size of the second‐largest component is Θ(?^‐2log n). Each of these results corresponds to an analogous well‐known result for the Erd?s‐Rényi process. Our proof techniques include combinatorial arguments, the differential equation method for random processes, and the singularity analysis of the moment generating function for the susceptibility, which satisfies a quasi‐linear partial differential equation. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

A Lower Bound of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> Norms in the CLT for Strongly Mixing Random Variables

We derive a lower bound of L _p norms, 1 ⩽ p ⩽ ∞, in the central limit theorem for strongly mixing random variables X ₁,..., X _n with under the boundedness condition ℙ{|X _i | ⩽ M} = 1 with a nonrandom constantM > 0 and condition ∑ _r⩾1 r ²α(r) < ∞, where α(r) are the Rosenblatt strong mixing coefficients. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 587–602, October–December, 2005.     

Lp-Approximable Sequences of Vectors and Limit Distribution of Quadratic Forms of Random Variables

The properties of L₂-approximable sequences established here form a complete toolkit for statistical results concerning weighted sums of random variables, where the weights are nonstochastic sequences approximated in some sense by square-integrable functions and the random variables are “two-wing” averages of martingale differences. The results constitute the first significant advancement in the theory of L₂-approximable sequences since 1976 when Moussatat introduced a narrower notion of L₂-generated sequences. The method relies on a study of certain linear operators in the spaces L_p and l_p. A criterion of L_p-approximability is given. The results are new even when the weight generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.     

Invariance principle for a class of non stationary processes with long memory

We prove a functional central limit theorem for the partial sums of a class of time varying processes with long memory. To cite this article: A. Philippe et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Analyticity of solutions of analytic non-linear general elliptic boundary value problems, and some results about linear problems

The main aim of this paper is to discuss the problem concerning the analyticity of the solutions of analytic non-linear elliptic boundary value problems. It is proved that if the corresponding first variation is regular in Lopatinskiĭ sense, then the solution is analytic up to the boundary. The method of proof really covers the case that the corresponding first variation is regularly elliptic in the sense of Douglis-Nirenberg-Volevich, and hence completely generalize the previous result of C. B. Morrey. The author also discusses linear elliptic boundary value problems for systems of elliptic partial differential equations where the boundary operators are allowed to have singular integral operators as their coefficients. Combining the standard Fourier transform technique with analytic continuation argument, the author constructs the Poisson and Green’s kernel matrices related to the problems discussed and hence obtain some representation formulae to the solutions. Some a priori estimates of Schauder type and L _p type are obtained. __________ Translated from Acta Sci. Natur. Univ. Jilin, 1963, (2): 403–447 by GAO Wenjie.     

Faisceaux\bar{\partial}-cohérents sur les variétés complexes

Résumé Dans ce travail nous généralisons au contexte des faisceaux analytiques cohérents un résultat classique de Koszul–Malgrange (Koszul and Malgrange in Arch Math 9 : 102–109, 1958) concernant l’intégrabilité des connexions de type (0,1) sur un fibré vectoriel complexe C ^∞ au-dessus d’une variété complexe. En introduisant la notion de faisceau -cohérent, qui est une notion qui vit dans le contexte C ^∞, nous montrons l’existence d’une équivalence (exacte) entre la catégorie des faisceaux analytiques cohérents et la catégorie des faisceaux -cohérents. Une application de notre caractérisation est une méthode (la -stabilité) qui permet de trouver des structures analytiques par déformation C ^∞ d’autres structures analytiques.     

On the tvGARCH(1,1) model: Existence, CLT, and tail index

We study the properties of the tvGARCH(1, 1) model (1.8) with logistic coefficients. We obtain conditions for the existence of an L ^p -bounded solution (p ⩾ 0) of Eq. (1.8). For p ⩾ 4, we prove an exponential decay of the lagged correlation function and the central limit theorem for partial sums of the squared tvGARCH(1, 1) process under similar conditions as in the stationary case. In the second part of the paper, we study the (weak) tail index of the tvGARCH(1, 1) model.     

Stable Grothendieck polynomials and <Emphasis Type="Italic">K</Emphasis>-theoretic factor sequences

We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as new proofs of the rule of Fomin and Greene (Discret Math 193:565–596, 1998) for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood–Richardson rule of Buch (Acta Math 189(1):37–78, 2002). The proof is based on a generalization of the Robinson–Schensted and Edelman–Greene insertion algorithms. Our results are applied to prove a number of new formulas and properties for K-theoretic quiver polynomials, and the Grothendieck polynomials of Lascoux and Schützenberger (C R Acad Sci Paris Ser I Math 294(13):447–450, 1982). In particular, we provide the first K-theoretic analogue of the factor sequence formula of Buch and Fulton (Invent Math 135(3):665–687, 1999) for the cohomological quiver polynomials.     

The convergence of the Rényi entropy of the normalized sums of IID random variables

We prove that the Rényi entropy of order $\alpha$ ($\alpha >1$) of the normalized sums of IID random variables with continuous differentiable density is convergent to the Rényi entropy of order $\alpha$ of the standard Gaussian distribution, and obtain the corresponding rates of convergence.     

A Mordell-Lang plus Bogomolov type result for curves in $${\mathbb{G}_{m}^{2}}$$

We prove a sharper so-called Mordell-Lang plus Bogomolov type result for curves lying in the two-dimensional linear torus. We mainly follow the approach of Rémond in (Comp Math 134:337–366, 2002), using Vojta and Mumford type inequalities. In the special case we consider, we improve Rémond’s main result using a better Bogomolov property and an elementary arithmetic Bézout theorem.      

On the intersection of two subgeometries of PG(n, q)

The study of the intersection of two Baer subgeometries of PG(n, q), q a square, started in Bose et al. (Utilitas Math 17, 65–77, 1980); Bruen (Arch Math 39(3), 285–288, (1982). Later, in Svéd (Baer subspaces in the n-dimensional projective space. Springer-Verlag) and Jagos et al. (Acta Sci Math 69(1–2), 419–429, 2003), the structure of the intersection of two Baer subgeometries of PG(n, q) has been completely determined. In this paper, generalizing the previous results, we determine all possible intersection configurations of any two subgeometries of PG(n, q).