Law of large numbers for weighted inductive means in a Hadamard space

We first study the law of large numbers for weighted inductive means of independent identically distributed random variables taking values in a Hadamard space. Secondly, we give a sharp asymptotic estimate for the limit of (non‐weighed) inductive means for the p‐Schatten class.     

Limit theorems for sums of partial quotients of continued fractions

We prove that the partial quotientsa _j of the regular continued fraction expansion cannot satisfy a strong law of large numbers for any reasonably growing norming sequence, and that thea _j belong to the domain of normal attraction to a stable law with characteristic exponent 1. We also show that thea _j satisfy a central limit theorem if a few of the largest ones are trimmed.In memory of Wilfried Nöbauer     

Strong approximations for Markovian service networks

Inspired by service systems such as telephone call centers, we develop limit theorems for a large class of stochastic service network models. They are a special family of nonstationary Markov processes where parameters like arrival and service rates, routing topologies for the network, and the number of servers at a given node are all functions of time as well as the current state of the system. Included in our modeling framework are networks of M _t /M _t /n _t queues with abandonment and retrials. The asymptotic limiting regime that we explore for these networks has a natural interpretation of scaling up the number of servers in response to a similar scaling up of the arrival rate for the customers. The individual service rates, however, are not scaled. We employ the theory of strong approximations to obtain functional strong laws of large numbers and functional central limit theorems for these networks. This gives us a tractable set of network fluid and diffusion approximations. A common theme for service network models with features like many servers, priorities, or abandonment is “non-smooth” state dependence that has not been covered systematically by previous work. We prove our central limit theorems in the presence of this non-smoothness by using a new notion of derivative. This revised version was published online in June 2006 with corrections to the Cover Date.     

Branching structure of uniform recursive trees

The branching structure of uniform recursive trees is investigated in this paper. Using the method of sums for a sequence of independent random variables, the distribution law of ηn, the number of branches of the uniform recursive tree of size n are given first. It is shown that the strong law of large numbers, the central limit theorem and the law of iterated logarithm for ηn follow easily from this method. Next it is shown that ηn and ξn, the depth of vertex n, have the same distribution, and the distribution law of ζn,m, the number of branches of size m, is also given, whose asymptotic distribution is the Poisson distribution with parameter λ= 1/m. In addition, the joint distribution and the asymptotic joint distribution of the numbers of various branches are given. Finally, it is proved that the size of the biggest branch tends to infinity almost sure as n→∞.     

A system of Markov processes with random lifetimes

Summary Let E be a locally compact Hausdorff space with a countable base, and suppose {x_n} is a countable collection of points in E. Particles enter E at the site x _n according to a Poisson process N _n (t). Upon entrance to E, a typical particle moves through the space, independently of all other particles, according to the transition law of a Markov process, until its death, which occurs at some random time D. We prove several limit theorems for various functional of this infinite particle system. In particular, laws of large numbers, and central limit theorems are proved for occupation times of relatively compact Borel sets.Supported in part by Arizona State University Grant-in-Aid     

A strong law of large numbers on the harmonic <Emphasis Type="Italic">p</Emphasis>-combination

In this paper, we establish a strong law of large numbers for the harmonic p-combinations of random star bodies. Starting from this theorem, we prove a strong law of large numbers in L _p space and provide the probabilistic version of dual Brunn-Minkowski inequality.     

Minimal subschemes of the group association schemes of Mathieu groups

Minimal subschemes of the group association schemes of Mathieu groupsM _n (n = 11, 12, 22, 23, 24) are determined. It is proved that for eachM _n (n = 11, 12, 22, 23, 24), there is a unique minimal subscheme of. The class numbers of these minimal subschemes are 7, 11, 9, 11 and 20 respectively. A general computer program to determine subschemes of group association schemes of relatively small class numbers is discussd.     

A family of non-oscillatory 6-point interpolatory subdivision schemes

In this paper we propose and analyze a new family of nonlinear subdivision schemes which can be considered non-oscillatory versions of the 6-point Deslauries-Dubuc (DD) interpolatory scheme, just as the Power _p schemes are considered nonlinear non-oscillatory versions of the 4-point DD interpolatory scheme. Their design principle may be related to that of the Power _p schemes and it is based on a weighted analog of the Power _p mean. We prove that the new schemes reproduce exactly polynomials of degree three and stay ’close’ to the 6-point DD scheme in smooth regions. In addition, we prove that the first and second difference schemes are well defined for each member of the family, which allows us to give a simple proof of the uniform convergence of these schemes and also to study their stability as in [19, 22]. However our theoretical study of stability is not conclusive and we perform a series of numerical experiments that seem to point out that only a few members of the new family of schemes are stable. On the other hand, extensive numerical testing reveals that, for smooth data, the approximation order and the regularity of the limit function may be similar to that of the 6-point DD scheme and larger than what is obtained with the Power _p schemes.     

Limit theorems for inverse process <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> of Hawkes process

We consider linear Hawkes process N _t and its inverse process T _n . The limit theorems for N _t are well known and studied by many authors. In this paper, we study the limit theorems for T _n . In particular, we investigate the law of large numbers, the central limit theorem and the large deviation principle for T _n . The main tool of the proof is based on immigration-birth representation and the observations on the relation between N _t and T _n .     

Convergence of weighted sums of random functions in D[0, 1]

Conditions are investigated which imply the tightness of certain weighted sums Σ_{i = 1}^k_n a_niX_i of random functions (X_n) taking values in D([0, 1]; E), where E is a separable Banach space. Improved weak laws of large numbers result as corollaries. Examples are presented to clarify the relative strengths of the moment conditions and their relationship to tightness and the strong law of large numbers. A tightness condition is defined using a certain class of sets measurable in the Skorokhod J₁-topology, which yields J₁-tightness of sequences of weighted sums. As a consequence, tightness of a sequence (X_n) in the Skorokhod M₁-topology is used to obtain J₁-tightness of a sequence ( ) of averages and a strong law of large numbers in D(R₊).     

Approximating Rough Stochastic PDEs

We study approximations to a class of vector‐valued equations of Burgers type driven by a multiplicative space‐time white noise. A solution theory for this class of equations has been developed recently in Probability Theory Related Fields by Hairer and Weber. The key idea was to use the theory of controlled rough paths to give definitions of weak/mild solutions and to set up a Picard iteration argument. In this article the limiting behavior of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Itô‐Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.© 2014 Wiley Periodicals, Inc.     

On the strong law of large numbers for sequences of random variables without the independence condition

New sufficient conditions for the applicability of the strong law of large numbers to a sequence of dependent random variables X ₁, X ₂, …, with finite variances are established. No particular type of dependence between the random variables in the sequence is assumed. The statement of the theorem involves the classical condition Σ _n ^∞ (log₂ n)²/n ² < ∞, which appears in various theorems on the strong law of large numbers for sequences of random variables without the independence condition.     

On Generalized Friedrichs and Krein‐von Neumann Extensions and Canonical Systems

Let S be a densely defined and closed symmetric relation in a Hilbert space ℋ︁ with defect numbers (1,1), and let A be some of its canonical selfadjoint extensions. According to Krein's formula, to S and A corresponds a so‐called Q‐function from the Nevanlinna class N . In this note we show to which subclasses N _γ of N the Q‐functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q‐functions of the generalized Friedrichs and Krein‐von Neumann extensions. A result of L. de Branges implies that to each function Q ∈ N there corresponds a unique Hamiltonian H such that Q is the Titchmarsh‐Weyl coefficient of the two‐dimensional canonical system J_y′ = —zHy on [0, ∞) where Weyl's limit point case prevails at ∞. Then the boundary condition y(0) = 0 corresponds to a symmetric relation T_min with defect numbers (1,1) in the Hilbert space L²_H, and Q is equal to the Q‐function with respect to the extension corresponding to the boundary condition y₁(0) = 0. If H satisfies some growth conditions at 0 or ∞, wepresent results on the corresponding Q‐functions and show under which conditions the generalized Friedrichs or Krein‐von Neumann extension exists.     

Fragments of Arithmetic and true sentences

By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the Π_n+1‐sentences true in the standard model is the only (up to deductive equivalence) consistent Π_n+1‐theory which extends the scheme of induction for parameter free Π_n+1‐formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first‐order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a consequence, we obtain results on the quantifier complexity, (non)finite axiomatizability and relative strength of schemes for Δ_n+1‐formulas. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Derivation of a quasi‐stationary coupled Darcy–Reynolds equation for incompressible viscous fluid flow through a thin porous medium with a fissure

We consider a non‐stationary Stokes system in a thin porous medium of thickness ε that is perforated by periodically distributed solid cylinders of size ε, and containing a fissure of width η_ε. Passing to the limit when ε goes to zero, we find a critical size in which the flow is described by a 2D quasi‐stationary Darcy law coupled with a 1D quasi‐stationary Reynolds problem. Copyright © 2017 John Wiley & Sons, Ltd.     

Laws of large numbers and the central limit theorem for sequences of coefficients of rotational expansions

For rotational expansions introduced in [1], conditions under which the law of large numbers, the strong law of large numbers, or the central limit theorem hold for Markov sequences of coefficients, are found. Answers are given in terms of the rate of growth of the quotients a_n. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 313–322. This work was carried out under the partial support of the International Scientific Foundation, grant MQV-000. Translated by N. A. Sidorov.     

Limit theorems for monotonic particle systems and sequential deposition

We prove spatial laws of large numbers and central limit theorems for the ultimate number of adsorbed particles in a large class of multidimensional random and cooperative sequential adsorption schemes on the lattice, and also for the Johnson–Mehl model of birth, linear growth and spatial exclusion in the continuum. The lattice result is also applicable to certain telecommunications networks. The proofs are based on a general law of large numbers and central limit theorem for sums of random variables determined by the restriction of a white noise process to large spatial regions.     

Critical behavior in inhomogeneous random graphs

We study the critical behavior of inhomogeneous random graphs in the so‐called rank‐1 case, where edges are present independently but with unequal edge occupation probabilities. The edge occupation probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter w_i, where w_i denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W. In this case, the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power‐law case, i.e., the case where \begin{align*}{\mathbb{P}}(W\geq k)\end{align*} is proportional to k^‐(τ‐1) for some power‐law exponent τ > 3, a property which is then inherited by the asymptotic degree distribution. We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when \begin{align*}{\mathbb{P}}(W > k) \leq ck^{-(\tau-1)}\end{align*} for all k ≥ 1 and some τ > 4 and c > 0, the largest critical connected component in a graph of size n is of order n^2/3, as it is for the critical Erd?s‐Rényi random graph. When, instead, \begin{align*}{\mathbb{P}}(W > k)=ck^{-(\tau-1)}(1+o(1))\end{align*} for k large and some τ∈(3,4) and c > 0, the largest critical connected component is of the much smaller order n^{(τ‐2)/(τ‐1)}. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 480–508, 2013     

Why organizational networks in reality do not show scale-free distributions

This paper discusses chain of command networks that are most likely to exhibit the scale-free (SF) property in organizational networks, explaining why organizational networks do not show SF distributions. We propose an evolving hierarchical tree network model without explicit preferential attachment. The model simulates several kinds of chain of command networks with the span of control ranging from extreme homogeneity to extreme heterogeneity. In addition to traditional degree distribution, a new kind of cumulative-outdegree distribution p(K _cum =k _cum ) is introduced and discussed that gives a probability that a randomly selected node has exactly k _cum children nodes. Theoretical analysis and simulation results show that even if the network size is large enough to meet the demand of large-scale networks, the SF property can emerge only when a hierarchical tree lies in two extreme situations: (1) the exact same span of control exists at all levels of an organization; (2) the node outdegree (i.e. span of control) distribution obeys a power-law distribution. The empirical investigations show that real organization networks are between the two extreme situations. This is why organizational networks in reality do not show an SF degree distribution or SF cumulative-outdegree distribution. This finding shows that the SF property is the consequence of extreme situations, even though it is very common in nature and in society. In fact, the SF property is of no value in the study of problems in organizations.     

On the weak laws of large numbers for arrays of random variables

In this paper we obtain weak laws of large numbers (WLLNs) for arrays of random variables under the uniform Cesàro-type condition. As corollary, we obtain the result of Hong and Oh [Hong, D. H., Oh, K. S., 1995. On the weak law of large numbers for arrays. Statist. Probab. Lett. 22, 55–57]. Furthermore, we obtain a WLLN for an L_p-mixingale array without the conditions that the mixingale is uniformly integrable and the L_p-mixingale numbers decay to zero at a special rate.