On the Number of Cutpoints of the Transient Nearest Neighbor Random Walk on the Line

We consider transient nearest neighbor random walks on the positive part of the real line. We give criteria for the finiteness of the number of cutpoints and strong cutpoints. Examples and open problems are presented.     

On Transience Conditions for Markov Chains and Random Walks

We prove a new transience criterion for Markov chains on an arbitrary state space and give a corollary for real-valued chains. We show by example that in the case of a homogeneous random walk with infinite mean the proposed sufficient conditions are close to those necessary. We give a new proof of the well-known criterion for finiteness of the supremum of a random walk.     

On Distributional Properties of Perpetuities

We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of p-moments, p>0, as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.      

On finiteness and continuity of shot noise processes

For a rather general class of stochastic processes induced by time-stationary and by event-stationary random marked point processes, respectively, conditions are given for the almost sure finiteness of these processes and for their continuous dependence on the underlying random marked point process.     

Passage times of random walks and Lévy processes across power law boundaries

We establish an integral test involving only the distribution of the increments of a random walk S which determines whether limsup _n→∞(S_n/n^κ) is almost surely zero, finite or infinite when 1/2<κ<1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller [9] concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of κ≥0. The results, and those of [9], are also extended to Lévy processes.This work is partially supported by ARC Grant DP0210572.     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Reaction-diffusion Equations

This paper dealswith non-autonomous fractional stochastic reaction-diffusion equations driven by multiplicative noise with s ∈ (0,1). We first present some conditions for estimating the boundedness of fractal dimension of a random invariant set. Then we establish the existence and uniqueness of tempered pullback random attractors. Finally, the finiteness of fractal dimension of the random attractors is proved.     

On the Bol Loops

In the present paper, the Bol loops and related groups are studied. We suggest a universal way to construct a Bol loop and find criteria for simplicity and finiteness of such loops.     

Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras

Algebras and Representation Theory - We investigate criteria for von-Neumann finiteness and reversibility in some classes of non-associative algebras. Types of algebras that are studied include...     

Existential monadic second order logic on random rooted trees

We address questions of logic and expressibility in the context of random rooted trees. Infiniteness of a rooted tree is not expressible as a first order sentence, but is expressible as an existential monadic second order sentence (EMSO). On the other hand, finiteness is not expressible as an EMSO. For a broad class of random tree models, including Galton–Watson trees with offspring distributions that have full support, we prove the stronger statement that finiteness does not agree up to a null set with any EMSO. We construct a finite tree and a non-null set of infinite trees that cannot be distinguished from each other by any EMSO of given parameters. This is proved via set-pebble Ehrenfeucht games (where an initial colouring round is followed by a given number of pebble rounds).     

A Finiteness Theorem Via the Mean Curvature Flow with Surgery

In this article, we use the recently developed mean curvature flow with surgery for 2-convex hypersurfaces to prove certain isotopy existence and finally extrinsic finiteness results (in the spirit of Cheeger’s finiteness theorem) for the space of 2-convex closed embedded hypersurfaces in \({\mathbb {R}}^{n+1}\).     

Anwendung von Ideal — Transformationen

A structure theorem for the maximal ideal transform of a complete local ring is used to get criteria for integralness and finiteness of ideal transforms. Further we give an elementary proof of the theorem of Mori — Nagata.     

Subcritical catalytic branching random walk with finite or infinite variance of offspring number

Subcritical catalytic branching random walk on the d-dimensional integer lattice is studied. New theorems concerning the asymptotic behavior of distributions of local particle numbers are established. To prove the results, different approaches are used, including the connection between fractional moments of random variables and fractional derivatives of their Laplace transforms. In the previous papers on this subject only supercritical and critical regimes were investigated under the assumptions of finiteness of the first moment of offspring number and finiteness of the variance of offspring number, respectively. In the present paper, for the offspring number in the subcritical regime, the finiteness of the moment of order 1 + δ is required where δ is some positive number.     

具有线性乘积白噪声的随机非自治吊桥方程的随机指数吸引子

本文考虑具有线性乘积白噪声的随机非自治吊桥方程长时间行为.首先,建立了所研究共圈系统的适定性;第二步,研究了该系统随机吸引子的存在性;第三步,当随机系数趋于0时,得到了随机吸引子的上半连续性;第四步,通过``迭代''法证明了随机吸引子在高正则空间中的正则性;最后,给出了该系统随机指数吸引子的存在性,同时得到了吸引子的有限分形维数.     

Strong Laws of Large Numbers for Sublinear Expectation under Controlled 1st Moment Condition

This paper deals with strong laws of large numbers for sublinear expectation under controlled 1st moment condition. For a sequence of independent random variables, the author obtains a strong law of large numbers under conditions that there is a control random variable whose 1st moment for sublinear expectation is finite. By discussing the relation between sublinear expectation and Choquet expectation, for a sequence of i.i.d random variables, the author illustrates that only the finiteness of uniform 1st moment for sublinear expectation cannot ensure the validity of the strong law of large numbers which in turn reveals that our result does make sense.     

Steepest Descent Method with Random Step Lengths

The paper studies the steepest descent method applied to the minimization of a twice continuously differentiable function. Under certain conditions, the random choice of the step length parameter, independent of the actual iteration, generates a process that is almost surely R-convergent for quadratic functions. The convergence properties of this random procedure are characterized based on the mean value function related to the distribution of the step length parameter. The distribution of the random step length, which guarantees the maximum asymptotic convergence rate independent of the detailed properties of the Hessian matrix of the minimized function, is found, and its uniqueness is proved. The asymptotic convergence rate of this optimally created random procedure is equal to the convergence rate of the Chebyshev polynomials method. Under practical conditions, the efficiency of the suggested random steepest descent method is degraded by numeric noise, particularly for ill-conditioned problems; furthermore, the asymptotic convergence rate is not achieved due to the finiteness of the realized calculations. The suggested random procedure is also applied to the minimization of a general non-quadratic function. An algorithm needed to estimate relevant bounds for the Hessian matrix spectrum is created. In certain cases, the random procedure may surpass the conjugate gradient method. Interesting results are achieved when minimizing functions having a large number of local minima. Preliminary results of numerical experiments show that some modifications of the presented basic method may significantly improve its properties.     

Intermittency generated by attracting and weakly repelling fixed points

Recently for a class of critically intermittent random systems a phase transition was found for the finiteness of the absolutely continuous invariant measure. The systems for which this result holds are characterized by the interplay between a superexponentially attracting fixed point and an exponentially repelling fixed point. In this article we consider a closely related family of random systems with exponentially fast attraction to and polynomially fast repulsion from two fixed points, and show that such a phase transition still exists. The method of the proof however is different and relies on the construction of a suitable invariant set for the transfer operator.     

On exact scaling log-infinitely divisible cascades

In this paper we extend some classical results valid for canonical multiplicative cascades to exact scaling log-infinitely divisible cascades. We present an alternative construction of exact scaling infinitely divisible cascades based on a family of cones whose geometry naturally induces the exact scaling property. We complete previous results on non-degeneracy and moments of positive orders obtained by Barral and Mandelbrot, and Bacry and Muzy: we provide a necessary and sufficient condition for the non-degeneracy of the limit measures of these cascades, as well as for the finiteness of moments of positive orders of their total mass, extending Kahane’s result for canonical cascades. Our main results are analogues to the results by Kahane and Guivarc’h regarding the asymptotic behavior of the right tail of the total mass. They come from a “non-independent” random difference equation satisfied by the total mass of the measures. The non-independent structure brings new difficulties to study the random difference equation, which we overcome thanks to Dirichlet’s multiple integral formula and Goldie’s implicit renewal theory. We also discuss the finiteness of moments of negative orders of the total mass, and some geometric properties of the support of the measure.     

Duality for Algebraic Linear Programming

A duality theory for algebraic linear (integer) programming (ALP) is developed which is of the same importance for linear (integer) programming with linear algebraic objectives as linear programming duality is for classical LP. In particular, optimality criteria for primal, primal-dual, and dual methods are given which generalize feasibility and complementarity criteria of classical LP. Strong duality results are given for special combinatorial problems. Further, the validity and finiteness of a primal simplex method based on a feasibility criterion are proved in the case of nondiscrete variables. In this case a strong duality result is shown.     

On a Lower Bound of the Uniform Distance in the Central Limit Theorem for ϕ-Mixing Random Variables

We derive a lower bound of the uniform distance in the central limit theorem for real -mixing random variables under the finiteness of the eighth moments of summands. The main result of the present paper generalizes the corresponding author"s result obtained in 1997 for m-dependent random variables to the case of -mixing random variables.     

Microscopic justification of the stochastic f-model of critical dynamics

We use a microscopic approach to derive the stochastic Langevin equations for a quantum Bose liquid near the phase transition to the superfluid state. These equations can be obtained in local form if either the finiteness of the system dimensions or the boundedness of the time interval under study is taken into account. We give the microscopic expressions for random forces in the stochastic equations and show that the white noise approximation holds only in the critical region. We prove that the model for properly describing the critical fluctuations is precisely the F-model.