Catalytic branching random walks in ℤ d with branching at the origin

A time-continuous branching random walk on the lattice ? ^d , d ≥ 1, is considered when the particles may produce offspring at the origin only. We assume that the underlying Markov random walk is homogeneous and symmetric, the process is initiated at moment t = 0 by a single particle located at the origin, and the average number of offspring produced at the origin is such that the corresponding branching random walk is critical. The asymptotic behavior of the survival probability of such a process at moment t → ∞ and the presence of at least one particle at the origin is studied. In addition, we obtain the asymptotic expansions for the expectation of the number of particles at the origin and prove Yaglom-type conditional limit theorems for the number of particles located at the origin and beyond at moment t.     

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.      

Combining empirical likelihood and generalized method of moments estimators: Asymptotics and higher order bias

This paper proposes an estimator combining empirical likelihood (EL) and the generalized method of moments (GMM) by allowing the sample average moment vector to deviate from zero and the sample weights to deviate from n⁻¹. The new estimator may be adjusted through free parameter δ∈(0,1) with GMM behavior attained as δ?0 and EL as δ?1. When the sample size is small and the number of moment conditions is large, the parameter space under which the EL estimator is defined may be restricted at or near the population parameter value. The support of the parameter space for the new estimator may be adjusted through δ. The new estimator performs well in Monte Carlo simulations.     

Principal resonance responses of SDOF systems with small fractional derivative damping under narrow-band random parametric excitation

The principal resonance responses of nonlinear single-degree-of-freedom (SDOF) systems with lightly fractional derivative damping of order α (0 < α < 1) subject to the narrow-band random parametric excitation are investigated. The method of multiple scales is developed to derive two first order stochastic differential equation of amplitude and phase, and then to examine the influences of fractional order and intensity of random excitation on the first-order and second-order moment. As an example, the stochastic Duffing oscillator with fractional derivative damping is considered. The effects of detuning frequency parameter, the intensity of random excitation and the fractional order derivative damping on stability are studied through the largest Lyapunov exponent. The corresponding theoretical results are well verified through direct numerical simulations. In addition, the phenomenon of stochastic jump is analyzed for parametric principal resonance responses via finite differential method. The stochastic jump phenomena indicates that the most probable motion is around the larger non-trivial branch of the amplitude response when the intensity of excitation is very small, and the probable motion of amplitude responses will move from the larger non-trivial branch to trivial branch with the increasing of the intensity of excitation. Such stochastic jump can be considered as bifurcation.     

A strong invariance principle for negatively associated random fields

In this paper we obtain a strong invariance principle for negatively associated random fields, under the assumptions that the field has a finite (2 + δ)th moment and the covariance coefficient u(n) exponentially decreases to 0. The main tools are the Berkes-Morrow multi-parameter blocking technique and the Csörg?-Révész quantile transform method.     

Non-hyperbolicity in random regular graphs and their traffic characteristics

In this paper we prove that random d-regular graphs with d ≥ 3 have traffic congestion of the order O(n log _d?1 ³ n) where n is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically δ-hyperbolic for any non-negative δ almost surely as n → ∞.     

Finite size Lyapunov exponent for some simple models of turbulence

Exact expressions for the finite size Lyapunov exponent λ(δ) are found and analyzed for several idealized models of turbulence in 1D and 2D. Among them are a random walk with discrete time and continuously distributed jumps and an isotropic Brownian flow in 2D also known as the Kraichnan flow. For the former a surprising fact is a δ⁻¹ scaling for intermediate values of δ in contrast to δ⁻² well known for a random walk in continuous time (Brownian flow) and for a simple random walk in discrete time. For the Kraichnan flow an exact relation is established between the scaling of λ(δ) and the scaling of relative dispersion in time.     

Character theory of symmetric groups, subgroup growth of Fuchsian groups, and random walks

We develop a number of statistical aspects of symmetric groups (mostly dealing with the distribution of cycles in various subsets of Sn), asymptotic properties of (ordinary) characters of symmetric groups, and estimates for the multiplicities of root number functions of these groups. As main applications, we present an estimate for the subgroup growth of an arbitrary Fuchsian group, a finiteness result for the number of Fuchsian presentations of such a group (resolving a long-standing problem of Roger Lyndon), as well as a proof of a well-known conjecture of Roichman concerning the mixing time of random walks on symmetric groups.     

Limit theorems for a supercritical branching process with immigration in a random environment

Let (Z_n) be a supercritical branching process with immigration in a random environment. Firstly, we prove that under a simple log moment condition on the offspring and immigration distributions, the naturally normalized population size W_n converges almost surely to a finite random variable W. Secondly, we show criterions for the non-degeneracy and for the existence of moments of the limit random variable W. Finally, we establish a central limit theorem, a large deviation principle and a moderate deviation principle about log Z_n.     

Expected Number of Real Roots for Random Linear Combinations of Orthogonal Polynomials Associated with Radial Weights

In this note, we obtain asymptotic expected number of real zeros for random polynomials of the form

$$f_{n}(z)=\sum\limits_{j=0}^{n}{a^{n}_{j}}{c^{n}_{j}}z^{j}$$

where \({a^{n}_{j}}\) are independent and identically distributed real random variables with bounded (2 + δ)th absolute moment and the deterministic numbers \({c^{n}_{j}}\) are normalizing constants for the monomials z ^j within a weighted L ²-space induced by a radial weight function satisfying suitable smoothness and growth conditions.

     

Divisibility properties of hyperharmonic numbers

We extend Wolstenholme’s theorem to hyperharmonic numbers. Then, we obtain infinitely many congruence classes for hyperharmonic numbers using combinatorial methods. In particular, we show that the numerator of any hyperharmonic number in its reduced fractional form is odd. Then we give quantitative estimates for the number of pairs (n, r) lying in a rectangle where the corresponding hyperharmonic number \({ h_n^{(r)} }\) is divisible by a given prime number p. We also provide p-adic value lower bounds for certain hyperharmonic numbers. It is an open problem that given a prime number p, there are only finitely many harmonic numbers h _n which are divisible by p. We show that if we go to the higher levels r ≥  2, there are infinitely many hyperharmonic numbers \({ h_n^{(r)} }\) which are divisible by p. We also prove a finiteness result which is effective.     

On the normal approximation of a binomial random sum

We present upper bounds of the L _s norms of the normal approximation for random sums of independent identically distributed random variables X ₁ , X ₂ , . . . with finite absolute moments of order 2 + δ, 0 < δ ≤ 1, where the number of summands N is a binomial random variable independent of the summands X ₁ , X ₂ , . . . . The upper bounds obtained are of order (E N) ^?δ/2 for all 1 ≤ s ≤ ∞.     

Correlation Inequalities and Applications to Vector-Valued Gaussian Random Variables and Fractional Brownian Motion

In this paper we extend certain correlation inequalities for vector-valued Gaussian random variables due to Kolmogorov and Rozanov. The inequalities are applied to sequences of Gaussian random variables and Gaussian processes. For sequences of Gaussian random variables satisfying a correlation assumption, we prove a Borel-Cantelli lemma, maximal inequalities and several laws of large numbers. This extends results of Be?ka and Ciesielski and of Hytönen and the author. In the second part of the paper we consider a certain class of vector-valued Gaussian processes which are α-Hölder continuous in p-th moment. For these processes we obtain Besov regularity of the paths of order α. We also obtain estimates for the moments in the Besov norm. In particular, the results are applied to vector-valued fractional Brownian motions. These results extend earlier work of Ciesielski, Kerkyacharian and Roynette and of Hytönen and the author.     

Convergence rates in the complete moment of moving-average processes

In this paper, we discuss precise asymptotics for a new kind of moment convergence of the moving-average process $X_k = \sum\limits_{i = - \infty }^\infty {a_{i + k} \varepsilon _i }$ , k ??1, where {?? _i : ??? < i < ??} is a doubly infinite sequence of independent identically distributed random variables with mean zero and the finiteness of variance, {?? _i : ??? < i < ??} is an absolutely summable sequence of real numbers, i.e., $\sum\limits_{i = - \infty }^\infty {\left| {a_i } \right| < \infty }$ .     

Flows over edge-disjoint mixed multipaths and applications

For improving reliability of communication in communication networks, where edges are subject to failure, Kishimoto [Reliable flow with failures in a network, IEEE Trans. Reliability, 46 (1997) 308-315] defined a δ-reliable flow, for a given source-sink pair of nodes, in a network for δ∈(0,1], where no edge carries a flow more than a fraction δ of the total flow in the network, and proved a max-flow min-cut theorem with cut-capacites defined suitably. Kishimoto and Takeuchi in [A method for obtaining δ-reliable flow in a network, IECCE Fundamentals E-81A (1998) 776-783] provided an efficient algorithm for finding such a flow.When (1/δ) is an integer, say q, Kishimoto and Takeuchi [On m-route flows in a network, IEICE Trans. J-76-A (1993) 1185-1200 (in Japanese)] introduced the notion of a q-path flow. Kishimoto [A method for obtaining the maximum multi-route flows in a network, Networks 27 (1996) 279-291] proved a max-flow min-cut theorem for q-path flow between a given source-sink pair (s,t) of nodes and provided a strongly polynomial algorithm for finding a q-path flow from s to t of maximum flow-value.In this paper, we extend the concept of q-path flow to any real number q?1. When q(=1/δ) is fractional, we show that this general q-path flow can be viewed as a sum of some ⌈q⌉-path flow and some ⌊q⌋-path flow. We discuss several applications of this results, which include a simpler proof and generalization of a known result on wavelength division multiplexing problem.Finally we present a strongly polynomial, combinatorial algorithm for synthesizing an undirected network with minimum sum of edge capacities that satisfies (non-simultaneously) specified minimum requirements of q-path flow-values between all pairs of nodes, for a given real number q?1.     

Cohomology of locally closed semi-algebraic subsets

Let k be a non-Archimedean field, let ? be a prime number distinct from the characteristic of the residue field of k. If χ is a separated k-scheme of finite type, Berkovich’s theory of germs allows to define étale ?-adic cohomology groups with compact support of locally closed semi-algebraic subsets of χ ^an . We prove that these vector spaces are finite dimensional continuous representations of the Galois group of k ^sep /k, and satisfy the usual long exact sequence and Künneth formula. This has been recently used by E. Hrushovski and F. Loeser in a paper about the monodromy of the Milnor fibration. In this statement, the main difficulty is the finiteness result, whose proof relies on a cohomological finiteness result for affinoid spaces, recently proved by V. Berkovich.     

Initial Value Problems of Fractional Order with Fractional Impulsive Conditions

In this paper we intend to accomplish two tasks firstly, we address some basic errors in several recent results involving impulsive fractional equations with the Caputo derivative, and, secondly, we study initial value problems for nonlinear differential equations with the Riemann–Liouville derivative of order 0 < α ≤ 1 and the Caputo derivatives of order 1 < δ < 2. In both cases, the corresponding fractional derivative of lower order is involved in the formulation of impulsive conditions.     

Sharp large deviations for sums of bounded from above random variables

We show large deviation expansions for sums of independent and bounded from above random variables. Our moderate deviation expansions are similar to those of Cram′er(1938), Bahadur and Ranga Rao(1960), and Sakhanenko(1991). In particular, our results extend Talagrand's inequality from bounded random variables to random variables having finite(2 + δ)-th moments, where δ∈(0, 1]. As a consequence,we obtain an improvement of Hoeffding's inequality. Applications to linear regression, self-normalized large deviations and t-statistic are also discussed.     

The 2D Boussinesq equations with vertical viscosity and vertical diffusivity

This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the Lr-norm of the vertical velocity v for any 1<r<∞ is globally bounded and that the L^∞-norm of v controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace δ(−Δ) for δ>0 would guarantee the global regularity of classical solutions.     

A note on exact convergence rate in the local limit theorem for a lattice branching random walk

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z. Denote by Z_n(z) the number of particles in the n-th generation in the model for each z ∈ Z. We derive the exact convergence rate in the local limit theorem for Z_n(z) assuming a condition like "EN(log N)~(1+λ) ∞" for the offspring distribution and a finite moment condition on the motion law. This complements the known results for the strongly non-lattice branching random walk on the real line and for the simple symmetric branching random walk on the integer lattice.