Almost sure limit theorem for the maximum of a classof quasi-stationary sequences简

This paper investigates the problem of almost sure limit theorem for the maximum of quasi-stationary sequence based on the result of Turkman and Walker. We prove an almost sure limit theorem for the maximum of a class of quasi-stationary sequence under weak dependence conditions of D(uk,un) and αtn,ln = O(log log n).(1+ε).     

Semigroups of Distributions with Linear Jacobi Parameters

We show that a convolution semigroup {?? _t } of measures has Jacobi parameters polynomial in the convolution parameter t if and only if the measures come from the Meixner class. Moreover, we prove the parallel result, in a more explicit way, for the free convolution and the free Meixner class. We then construct the class of measures satisfying the same property for the two-state free convolution. This class of two-state free convolution semigroups has not been considered explicitly before. We show that it also has Meixner-type properties. Specifically, it contains the analogs of the normal, Poisson, and binomial distributions, has a Laha?CLukacs-type characterization, and is related to the q=0 case of quadratic harnesses.     

Vector-valued almost convergence and classical properties in normed spaces

In this paper we study the almost convergence and the almost summability in normed spaces. Among other things, spaces of sequences defined by the almost convergence and the almost summability are proved to be complete if the basis normed space is so. Finally, some classical properties such as completeness, reflexivity, Schur property, Grothendieck property, and the property of containing a copy of c ₀ are characterized in terms of the almost convergence.     

Almost sure central limit theory for products of sums of partial sums

Consider a sequence of i.i.d. positive random variables. An universal result in almost sure limit theorem for products of sums of partial sums is established.We will show that the almost sure limit the...     

Limit Property for Regular and Weak Generalized Convolution

We denote by ? \((\mathcal{P_{+}})\) the set of all probability measures defined on the Borel subsets of the real line (the positive half-line [0,∞)). K. Urbanik defined the generalized convolution as a commutative and associative ?₊-valued binary operation ? on ? ₊ ² which is continuous in each variable separately. This convolution is distributive with respect to convex combinations and scale changes T _a (a>0) with δ ₀ as the unit element. The key axiom of a generalized convolution is the following: there exist norming constants c _n and a measure ν other than δ ₀ such that \(T_{c_{n}}\delta_{1}^{\bullet n}\to\nu\).In Sect. 2 we discuss basic properties of the generalized convolution on ? which hold for the convolutions without the key axiom. This rather technical discussion is important for the weak generalized convolution where the key axiom is not a natural assumption. In Sect. 4 we show that if the weak generalized convolution defined by a weakly stable measure μ has this property, then μ is a factor of strictly stable distribution.     

Random Steiner symmetrizations of sets and functions

In this article we prove almost sure convergence, in the L ₁ distance, of sequences of random Steiner symmetrizations of measurable sets having finite measure to the ball having the same measure. From this result we deduce analogous statements concerning the almost sure convergence to the spherical symmetrization of random Steiner symmetrizations of non negative L _p functions in the natural norm and uniform convergence of non negative continuous functions with bounded support. The latter result is finally used to prove that sequences of random symmetrizations of a compact set converge almost surely in the Hausdorff distance to the ball having the same measure, providing another proof of Mani-Levitska’s conjecture besides the one given in 2006 by Van Schaftingen (Topol Methods Nonlinear Anal 28(1): 61–85, 2006).     

Approximate Solution of Nonlinear Discrete Equations of Convolution Type

By the method of potential monotone operators we prove global theorems on existence, uniqueness, and ways to find a solution for different classes of nonlinear discrete equations of convolution type with kernels of special form both in weighted and in weightless real spaces ? _p . Using the property of potentiality of the operators under consideration, in the case of space ? ₂ and in the case of a weighted space ? _p (?) with a generic weight ?, we prove that a discrete equation of convolution type with an odd power nonlinearity has a unique solution and it (the main result) can be found by gradient method.     

Almost Sure Convergence of the Kaczmarz Algorithm with Random Measurements

The Kaczmarz algorithm is an iterative method for reconstructing a signal x??? ^d from an overcomplete collection of linear measurements y _n =??x,?? _n ??, n??1. We prove quantitative bounds on the rate of almost sure exponential convergence in the Kaczmarz algorithm for suitable classes of random measurement vectors $\{\varphi_{n}\}_{n=1}^{\infty} \subset {\mathbb {R}}^{d}$ . Refined convergence results are given for the special case when each ?? _n has i.i.d. Gaussian entries and, more generally, when each ?? _n /???? _n ?? is uniformly distributed on $\mathbb{S}^{d-1}$ . This work on almost sure convergence complements the mean squared error analysis of Strohmer and Vershynin for randomized versions of the Kaczmarz algorithm.     

Multidimensional integral operators with homogeneous kernels of compact type and multiplicatively weakly oscillating coefficients

In the space L _p (? ⁿ ), 1 < p < +∞, we consider a new class of integral operators with kernels homogeneous of degree ?n, which includes the class of operators with homogeneous SO(n)-invariant kernels; we study the Banach algebra generated by such operators with multiplicatively weakly oscillating coefficients. For operators from this algebra, we define a symbol in terms of which we formulate a Fredholm property criterion and derive a formula for calculating the index. An important stage in obtaining these results is the establishment of the relationship between the operators of the class under study and the operators of one-dimensional convolution with weakly oscillating compact coefficients.     

Almost sure convergence and bounded entropy

It is shown that the almost sure convergence property for certain sequences of operators {S _n{ implies a uniform bound on the metrical entropy of the sets {S _nf|n=1, 2, ...{, wheref is taken in theL ²-unit ball. This criterion permits one to unify certain counterexamples due to W. Rudin [Ru] and J.M. Marstrand [Mar] and has further applications. The theory of Gaussian processes is crucial in our approach.     

Generalization of Berg-Dimovski convolution in spaces of analytic functions

In the space H(G) of functions analytic in a ρ-convex region G equipped with the topology of compact convergence, we construct a convolution for the operator J _π+L where J _ρ is the generalized Gel’fond-Leont’ev integration operator and L is a linear continuous functional on H(G). This convolution is a generalization of the well-known Berg-Dimovski convolution. We describe the commutant of the operator J _π+L in ?(G) and obtain the representation of the coefficient multipliers of expansions of analytic functions in the system of Mittag-Leffler functions.     

Geometric Property （T）

This paper discusses“geometric property （T）”. This is a property of metric spaces introduced in earlier works of the authors for its applications to K-theory. Geometric property （T） is a strong form of “expansion property”, in particular, for a sequence （Xn） of bounded degree finite graphs, it is strictly stronger than （Xn） being an expander in the sense that the Cheeger constants h（Xn） are bounded below. In this paper, the authors show that geometric property （T） is a coarse invariant, i.e., it depends only on the large-scale geometry of a metric space X. The authors also discuss how geometric property （T） interacts with amenability, property （T） for groups, and coarse geometric notions of a-T-menability. In particular, it is shown that property （T） for a residually finite group is characterised by geometric property （T） for its finite quotients.     

The smallest graphs with certain adjacency properties

A graph is said to have property P₁,_n if for every sequence of n + 1 points, there is another point adjacent only to the first point. It has previously been shown that almost all graphs have property P₁,_n. It is easy to verify that for each n, there is a cube with this property. A more delicate question asks for the construction of the smallest graphs having property P₁,_n. We find that this problem is intimately related with the discovery of the highly symmetric graphs known as cages, and are thereby enabled to resolve this question for 1?n?6.     

Lacunary systems and generalized linear processes

Herein we obtain several almost sure and L_p convergence properties of generalized linear processes. When the generalized linear process is generated by an orthonormal S_p system (which includes the classical i.i.d. white noise, L_p bounded martingale difference sequences and weakly multiplicative sequences as special cases), it is shown that these convergence properties are related to certain second order properties of the process.     

An almost sure central limit theorem for self-normalized products of sums of i.i.d. random variables

Let X,X₁,X₂,… be a sequence of independent and identically distributed positive random variables with EX=μ>0. In this paper we show that the almost sure central limit theorem for self-normalized products of sums holds only under the assumptions that X belongs to the domain of attraction of the normal law.     

The Second Appell Function for one Large Variable

We consider the Mellin convolution integral representation of the second Appell function given in [8]. Then, we apply the asymptotic method designed in [12] for this kind of integrals to derive new asymptotic expansions of the Appell function F ₂ for one large variable in terms of hypergeometric functions. For certain values of the parameters, some of these expansions involve logarithmic terms in the asymptotic variables. The accuracy of the approximations is illustrated with numerical experiments.     

Stochastic functional evolution equations with monotone nonlinearity: Existence and stability of the mild solutions

In this paper, we study a class of semilinear functional evolution equations in which the nonlinearity is demicontinuous and satisfies a semimonotone condition. We prove the existence, uniqueness and exponentially asymptotic stability of the mild solutions. Our approach is to apply a convenient version of Burkholder inequality for convolution integrals and an iteration method based on the existence and measurability results for the functional integral equations in Hilbert spaces. An Itô-type inequality is the main tool to study the uniqueness, p-th moment and almost sure sample path asymptotic stability of the mild solutions. We also give some examples to illustrate the applications of the theorems and meanwhile we compare the results obtained in this paper with some others appeared in the literature.     

Ratio limit theorems for branching Ornstein-Uhlenbeck processes

We prove ratio limit theorems for critical ano supercritical branching Ornstein-Uhlenbeck processes. A finite first moment of the offspring distribution {p_n} assures convergence in probability for supercritical processes and conditional convergence in probability for critical processes. If even Σp_nnlog⁺log⁺n< ∞, then almost sure convergence obtains in the supercritical case.     

Extended Kac-Akhiezer formulae and the Fredholm determinant of finite section Hilbert-Schmidt kernels

This paper deals with some results (known as Kac-Akhiezer formulae) on generalized Fredholm determinants for Hilbert-Schmidt operators onL ₂-spaces, available in the literature for convolution kernels on intervals. The Kac-Akhiezer formulae have been obtained for kernels, which are not necessarily of convolution nature and for domains in ? ⁿ .     

Right Markov Processes and Systems of Semilinear Equations with Measure Data

In the paper we prove the existence of probabilistic solutions to systems of the form ?Au = F(x, u) + μ, where F satisfies a generalized sign condition and μ is a smooth measure. As for A we assume that it is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on L¹. This class includes local and nonlocal operators corresponding to Dirichlet forms as well as some operators which are not in the variational form. To study the problem we introduce new concept of compactness property relating the underlying Markov process to almost everywhere convergence. We prove some useful properties of the compactness property and provide its characterization in terms of Meyer’s property (L) of Markov processes and in terms of order compactness of the associated resolvent.