On the Existence of <Emphasis Type="Italic">φ</Emphasis>-Moments of the Limit of a Normalized Supercritical Galton–Watson Process

Let (Z _n ) _{n 0} be a supercritical Galton–Watson process with finite re-production mean  and normalized limit W=lim _n ^–n Z _n . Let further : [0,) [0,) be a convex differentiable function with (0)=(0)=0 and such that ( ) is convex with concave derivative for some n 0. By using convex function inequalities due to Topchii and Vatutin, and Burkholder, Davis and Gundy, we prove that 0 < E (W) < if, and only if, , where

A Question in the Theory of Totally Local Formations of Finite Groups

It is proved that all proper totally local subformations of a non one-generated totally local formation of finite groups are one-generated iff coincides with a formation of all soluble -groups, where ||=2.     

On the expansion of the resolvent for elliptic boundary contact problems

Let A be an elliptic operator on a compact manifold with boundary , and let be a covering map, where Y is a closed manifold. Let A _C be a realization of A subject to a coupling condition C that is elliptic with parameter in the sector Λ. By a coupling condition we mean a nonlocal boundary condition that respects the covering structure of the boundary. We prove that the resolvent trace for N sufficiently large has a complete asymptotic expansion as . In particular, the heat trace has a complete asymptotic expansion as , and the -function has a meromorphic extension to .      

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.      

On the Weak Limiting Behavior of Almost Surely Convergent Row Sums from Infinite Arrays of Rowwise Independent Random Elements in Banach Spaces

For an array {V _nk ,k1,n1} of rowwise independent random elements in a real separable Banach space with almost surely convergent row sums , we provide criteria for S _n –A _n to be stochastically bounded or for the weak law of large numbers to hold where {A _n ,n1} is a (nonrandom) sequence in .     

Rates of Decay and h-Processes for One Dimensional Diffusions Conditioned on Non-Absorption

Let (X _t ) be a one dimensional diffusion corresponding to the operator , starting from x>0 and T ₀ be the hitting time of 0. Consider the family of positive solutions of the equation with (0, ), where . We show that the distribution of the h-process induced by any such is , for a suitable sequence of stopping times (S _M : M0) related to which converges to with M. We also give analytical conditions for , where is the smallest point of increase of the spectral measure associated to .     

A Discrete Limit Theorem for the Matsumoto Zeta‐Function in the Space of Analytic Functions

In the paper, a discrete distribution of the Matsumoto zetafunction is considered. It is proved that the probability measure , converges weakly as .     

On Distance-Regular Graphs with Height Two, II

Let be a distance-regular graph with diameter and height , where . Suppose that for every in and every in , the induced subgraph on is isomorphic to a complete multipartite graph with . Then and is isomorphic to the Johnson graph .     

Nontrivial Phase Transition in a Continuum Mirror Model

We consider a Poisson point process on with intensity , and at each Poisson point we place a two sided mirror of random length and orientation. The length and orientation of a mirror is taken from a fixed distribution, and is independent of the lengths and orientations of the other mirrors. We ask if light shone from the origin will remain in a bounded region. We find that there exists a with 0 < < for which, if < , light leaving the origin in all but a countable number of directions will travel arbitrariliy far from the origin with positive probability. Also, if > , light from the origin will almost surely remain in a bounded region.     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

A Change-of-Variable Formula with Local Time on Curves

Let be a continuous semimartingale and let be a continuous function of bounded variation. Setting and suppose that a continuous function is given such that F is C^1,2 on and F is on . Then the following change-of-variable formula holds: where is the local time of X at the curve b given by and refers to the integration with respect to . A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping.     

Limit theorems for compact two-point homogeneous spaces of large dimensions

Let be the field , , or of real dimension . For each dimensiond2, we study isotropic random walks(Y ₁)₁0 on the projective space with natural metricD where the random walk starts at some with jumps at each step of a size depending ond. Then the random variablesX ₁ ^d :=cosD(Y ₁ ^d ,x ₀ ^d ) form a Markov chain on [–1, 1] whose transition probabilities are related to Jacobi convolutions on [–1, 1]. We prove that, ford, the random variables (vd/2)(X _l(d) ^d +1) tend in distribution to a noncentral ²-distribution where the noncentrality parameter depends on relations between the numbers of steps and the jump sizes. We also derive another limit theorem for as well as thed-spheresS ^d ford.     

Lattices of interpretability types of varieties

Let be the set of all primes, the field of all algebraic numbers, and Z the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as , and , where AD is a variety of -divisible Abelian groups with unique taking of the pth root _p(x) for every p , is a variety of -modules over a normal field , contained in , and G_n is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that , and are distributive lattices, with and where ub and ub_f are lattices (w.r.t. inclusion) of all subsets of the set and of finite subsets of , respectively.Deceased.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.     

Selective Limit Theorems for Random Walks on Parabolic Biangle and Triangle Hypergroups

Let K be respectively the parabolic biangle and the triangle in and be a sequence in [0, +[ such that lim_p (p)=+. According to Koornwinder and Schwartz,⁽⁷⁾ for each there exist a convolution structure (*_(p)) such that (K, *_(p)) is a commutative hypergroup. Consider now a random walk on (K, *_(p)), assume that this random walk is stopped after j(p) steps. Then under certain conditions given below we prove that the random variables on K admit a selective limit theorems. The proofs depend on limit relations between the characters of these hypergroups and Laguerre polynomials that we give in this work.     

Uniform and universal Glivenko-Cantelli classes

A class of measurable functions on a probability space is called a Glivenko-Cantelli class if the empirical measuresP _n converge to the trueP uniformly over almost surely. is a universal Glivenko-Cantelli class if it is a Glivenko-Cantelli Cantelli class for all lawsP on a measurable space, and a uniform Glivenko-Cantelli class if the convergence is also uniform inP. We give general sufficient conditions for the Glivenko-Cantelli and universal Glivenko-Cantelli properties and examples to show that some stronger conditions are not necessary. The uniform Glivenko-Cantelli property is characterized, under measurability assumptions, by an entropy condition.     

Diffeomorphisms of the Circle and the Beurling–Helson Theorem

We consider the algebra of absolutely convergent Fourier series on the circle . According to the Beurling–Helson theorem, the condition , implies that is trivial: . We construct a nontrivial diffeomorphism of onto itself such that , where (n) is an arbitrary given sequence with . By analogy with a conjecture due to Kahane, it is natural to suppose that this rate of growth is the slowest possible.     

Rates of convergence for the stability of large order statistics

Forr1 and eachnr, letM _nr be therth largest ofX ₁,X ₂, ...,X _n , where {X _n ,n1} is an i.i.d. sequence. Necessary and sufficient conditions are presented for the convergence of for all >0 and some –1, where {a _n } is a real sequence. Furthermore, it is shown that this series converges for all >–1, allr1 and all >0 if it converges for some >–1, somer1 and all >0.     

A Slow Transient Diffusion in a Drifted Stable Potential

We consider a diffusion process X in a random potential of the form , where is a positive drift and is a strictly stable process of index with positive jumps. Then the diffusion is transient and converges in law towards an exponential distribution. This behaviour contrasts with the case where is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as “slow” as in the recurrent setting.      

On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

A result by Elton⁽⁶⁾ states that an iterated function system

Extremal stochastic integrals: a parallel between max-stable processes and α-stable processes

We construct extremal stochastic integrals of a deterministic function with respect to a random Fréchet () sup-measure. The measure is sup-additive rather than additive and is defined over a general measure space , where is a deterministic control measure. The extremal integral is constructed in a way similar to the usual stable integral, but with the maxima replacing the operation of summation. It is well-defined for arbitrary , and the metric metrizes the convergence in probability of the resulting integrals.This approach complements the well-known de Haan's spectral representation of max-stable processes with Fréchet marginals. De Haan's representation can be viewed as the max-stable analog of the LePage series representation of stable processes, whereas the extremal integrals correspond to the usual stable stochastic integrals. We prove that essentially any strictly stable process belongs to the domain of max-stable attraction of an Fréchet, max-stable process. Moreover, we express the corresponding Fréchet processes in terms of extremal stochastic integrals, involving the kernel function of the stable process. The close correspondence between the max-stable and stable frameworks yields new examples of max-stable processes with non-trivial dependence structures.This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grant DMS-0505747 at Boston University.