Random coin tossing

Summary. A sequence of heads and tails is produced by repeatedly selecting a coin from two possible coins, and tossing it. The second coin is tossed at renewal times in a renewal process, and the first coin is tossed at all other times. The first coin is fair (Prob(heads)=1/2), and the second coin is known either to be fair, or to have known biasθ∈(0,1] (Prob(heads) ). Letting u _k := Prob (There is a renewal at time k), we show that if ∑ _{k =0} ^∞ u _k ²=∞, we can determine, using only the sequence of heads and tails produced, if the second coin had bias θ or 0. If , we show that this is not possible. Received: 20 November 1996 / In revised form: 20 February 1997     

M-Estimation for Discretely Observed Ergodic Diffusion Processes with Infinitely Many Jumps

We study parametric inference for multidimensional stochastic differential equations with jumps from some discrete observations. We consider a case where the structure of jumps is mainly controlled by a random measure which is generated by a Lévy process with a Lévy measure f_θ(z)dz, and we admit the case ∫ f_θ(z)dz = ∞ in which infinitely many small jumps occur even in any finite time intervals. We propose an estimating function under this complicated situation and show the consistency and the asymptotic normality. Although the estimators in this paper are not completely efficient, the method can be applied to comparatively wide class of stochastic differential equations, and it is easy to compute the estimating equations. Therefore, it may be useful in applications. We also present some simulation results for some simple models. Final version 25 December 2004     

Two explicit Skorokhod embeddings for simple symmetric random walk

Motivated by problems in behavioural finance, we provide two explicit constructions of a randomized stopping time which embeds a given centred distribution $\mu$ on integers into a simple symmetric random walk in a uniformly integrable manner. Our first construction has a simple Markovian structure: at each step, we stop if an independent coin with a state-dependent bias returns tails. Our second construction is a discrete analogue of the celebrated Azéma–Yor solution and requires independent coin tosses only when excursions away from maximum breach predefined levels. Further, this construction maximizes the distribution of the stopped running maximum among all uniformly integrable embeddings of $\mu$.     

Countably generated prime ideals in <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>

We confirm a twenty year old conjecture by showing that a nonzero prime ideal P in the algebra H^∞ of bounded analytic functions in the open unit disk is countably generated if and only if it is either a principal ideal generated by the polynomial z−z₀, |z₀|<1, or if P is generated by the n-th roots of an atomic inner function. The case of the algebra H^∞+C is also dealt with. Dedicated to the 70th birthday of Joseph Cima Research supported by the RIP-program Oberwolfach 2004.     

The Region of Values of the System {f(z 1),..., f(z n )} in the Class of Typically Real Functions

The paper studies the region of values of the system {f(z₁), f(z₂),... , f(z_n)} in the class T of functions f(z) = z + a₂z² + ⋯ regular in the unit disk and satisfying the condition Im f(z) Im z > 0 for Im z ≠ 0. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 41–51.     

Solutions with boundary‐layers and spike‐layers to singularly perturbed quasilinear Dirichlet problems

The structure of nontrivial nonnegative solutions to singularly perturbed quasilinear Dirichlet problems of the form –?Δ_pu = f(u) in Ω, u = 0 on ?Ω, Ω ? R ^N a bounded smooth domain, is studied as ? → 0⁺, for a class of nonlinearities f(u) satisfying f(0) = f(z₁) = f(z₂) = 0 with 0 < z₁ < z₂, f < 0 in (0, z₁), f > 0 in (z₁, z₂) and f(u)/u^p–1 = –∞. It is shown that there are many nontrivial nonnegative solutions with spike‐layers. Moreover, the measure of each spike‐layer is estimated as ? → 0⁺. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0,∞). Uniqueness of a solution with a boundary‐layer and many positive intermediate solutions with spike‐layers are obtained for ? sufficiently small. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Randomness-optimal oblivious sampling

We present the first efficient oblivious sampler that uses an optimal number of random bits, up to an arbitrary constant factor bigger than 1. Specifically, for any α>0, it uses (1+α)(m+log γ⁻¹) random bits to output d=poly(ϵ⁻¹, log γ⁻¹, m) sample points z₁,…,z_d∈{0, 1}^m such that for any function f: {0, 1}^m→[0, 1], Pr [|(1/d)∑_i=1^df(z_i)− E f|≤ϵ]≥1−γ. Our proof is based on an improved extractor construction. An extractor is a procedure which takes as input the output of a defective random source and a small number of truly random bits, and outputs a nearly random string. We present the first optimal extractor, up to constant factors, for defective random sources with constant entropy rate. We give applications to constructive leader election and reducing randomness in interactive proofs. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 345–367 (1997)     

Moments of Absorption Time for a Conditioned Random Walk

A random walk on the set of integers {0,1,2,...,a} with absorbing barriers at 0 and a is considered. The transition times from the integers z (0<z<a) are random variables with finite moments. The nth moment of the time to absorption at a, D_z,n, conditioned on the walk starting at z and being absorbed at a, is discussed, and a difference equation with boundary values and initial values for D_z,n is given. It is solved in several special cases. The problem is motivated by questions from biology about tumor growth and multigene evolution which are discussed.     

Fast graphs for the random walker

 Consider the time T _oz when the random walk on a weighted graph started at the vertex o first hits the vertex set z. We present lower bounds for T _oz in terms of the volume of z and the graph distance between o and z. The bounds are for expected value and large deviations, and are asymptotically sharp. We deduce rate of escape results for random walks on infinite graphs of exponential or polynomial growth, and resolve a conjecture of Benjamini and Peres. Received: 31 October 2000 / Revised version: 5 January 2002 / Published online: 22 August 2002     

拟圆周的两个几何性质

§1　IntroductionLetΓbe a Jordan curve of R2 and f∶R2→R2 be a k-quasiconformal mapping,where1≤k<+∞.Γis called a quasicirlce ifΓis the image of the unit circle B2 under f.It is well-known that quasicircles play a very important role in quasiconformalmapping theory,complex dynamics,Fuchsian groups,Teichmuller space theory and lowdimensional topology,( see[1—5] etc.)In1 963 ,Ahlfors obtained the three-point property of quasidisks[6] .Later,Gehring[7] ,Osgood[8] ,Krzyz[9] ,Ch…     

The nth Order Implicit Differentiation Formula for Two Variables with an Application to Computing All Roots of a Transcendental Function

We prove a version of a generalization of the Lagrange inversion formula (LIF) for an implicit equation G(z, w) = 0 of two variables, expressing the nth derivative of z with respect to w as a polynomial in the mixed partial derivatives function of G with respect to z and w, and negative powers of the separant G_z o \frac?G?z{G_z \equiv \frac{\partial G}{\partial z}}. Our method of proof is original, using only induction, and hence requires only that G be n-times differentiable in both variables, and requires only that the separant be nonzero. We then move on to a novel application of this LIF-like formula to derive a power series formula for each of the countably infinitely many roots of a pseudopolynomial—a finite sum of powers of a variable but allowing the powers to be any complex numbers.     

Von Neumann and Newman poker with a flip of hand values

The von Neumann and Newman poker models are simplified two-person poker models in which hands are modeled by real values drawn uniformly at random from the unit interval. We analyze a simple extension of both models that introduces an element of uncertainty about the final strength of each player’s own hand, as is present in real poker games. Whenever a showdown occurs, an unfair coin with fixed bias q is tossed, 0≤q≤1/2. With probability 1−q, the higher hand value wins as usual, but, with the remaining probability q, the lower hand wins. Both models favor the first player for q=0 and are fair for q=1/2. Our somewhat surprising result is that the first player’s expected payoff increases with q as long as q is not too large. That is, the first player can exploit the additional uncertainty introduced by the coin toss and extract even more value from his opponent.     

On the clustering of independent uniform random variables

We consider the number K_n of clusters at a distance level d_n ∈ (0, 1) of n independent random variables uniformly distributed in [0, 1], or the number K_n of connected components in the random interval graph generated by these variables and d_n, and, depending upon how fast d_n → 0 as n → ∞, determine the asymptotic distribution of K_n, with rates of convergence, and of related random variables that describe the cluster sizes. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

SIR epidemics on random graphs with a fixed degree sequence

Let Δ > 1 be a fixed positive integer. For \begin{align*}{\textbf{ {z}}} \in \mathbb{R}_+^\Delta\end{align*} let G_z be chosen uniformly at random from the collection of graphs on ∥z∥₁n vertices that have z_in vertices of degree i for i = 1,…,Δ. We determine the likely evolution in continuous time of the SIR model for the spread of an infectious disease on G_z, starting from a single infected node. Either the disease halts after infecting only a small number of nodes, or an epidemic spreads to infect a linear number of nodes. Conditioning on the event that more than a small number of nodes are infected, the epidemic is likely to follow a trajectory given by the solution of an associated system of ordinary differential equations. These results also give the likely number of nodes infected during the course of the epidemic and the likely length in time of the epidemic. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

A Universality Property of Gaussian Analytic Functions

We consider random analytic functions defined on the unit disk of the complex plane f(z) = ?_n=0^￥ a_n X_n zⁿf(z) = \sum_{n=0}^{\infty} a_{n} X_{n} z^{n}, where the X _n ’s are i.i.d., complex-valued random variables with mean zero and unit variance. The coefficients a _n are chosen so that f(z) is defined on a domain of ℂ carrying a planar or hyperbolic geometry, and Ef(z)[`(f(w))]\mathbf{E}f(z)\overline{f(w)} is covariant with respect to the isometry group. The corresponding Gaussian analytic functions have been much studied, and their zero sets have been considered in detail in a monograph by Hough, Krishnapur, Peres, and Virág. We show that for non-Gaussian coefficients, the zero set converges in distribution to that of the Gaussian analytic functions as one transports isometrically to the boundary of the domain. The proof is elementary and general.     

Goodisman's correction of the Thomas-Fermi-von Weizsäcker theory of atoms

Summary Following a suggestion of J. Goodisman, we substitute the therm by in the Thomas-Fermi-von Weizsäcker energy functional for atoms.f _z³ [0,1] is a function depending on the nuclear chargez.We establish conditions for the functionsf _z such that the ratio of this modified TFW-energyE _kz ^/TFWG (z) (kz is the total number of electrons) and the exact quantum mechanical energy converges to 1 asz. Moreover, we prove thatE _kz ^/TFWG (z)=E _kz ^/TFW (z)+Dz ²+o(z ²) (z) and determineD. Here,E _kz ^/TFW (z) is the unmodified TFW energy.

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Zusammenfassung Einem Vorschlag J. Goodismans folgend, ersetzen wir in dem Thomas-Fermi-von Weizsäcker Energiefunktional für Atome den Term durch .f _z³ [0,1] ist hierbei eine Funktion, die von der Kernladungszahlz abhängt.Wir geben dann Bedingungen für die Funktionenf _z an, unter denen der Quotient der so modifizierten TFW-energieE _kz ^/TFWG (z) (kz=Anzahl der Elektronen) und der exakten quantenmechanischen Energie fürz gegen 1 konvergiert. Darüber hinaus beweisen wir, daßE _kz ^/TFWG (z)=E _kz ^/TFW (z)+D·o(itz) ² (z) gilt und bestimmmenD. E _kz ^/TFW (z) ist hierbei die nicht-modifizierte TFW-Energie.

     

Inverse problem for a perturbed stratified strip in two dimensions

We consider the divergence form elliptic operator A=??_x,z·(c²(x,z) ?_x,z) in the strip Ω=?× [0,H]. The velocity c(x,z) describes the multistratification of Ω: a horizontal stratification with a compact perturbation K, the velocity in K is a L^∞(K) function. We suppose that the position of the perturbation is known and we prove uniqueness for identification of the perturbation from one generalized eigenfunction pattern in the neighbourhood of K. Copyright © 2004 John Wiley & Sons, Ltd.     

Exponential Convergence in Probability for Empirical Means of Brownian Motion and of Random Walks

Given a Brownian motion (B _t) _t0 in R ^d and a measurable real function f on R ^d belonging to the Kato class, we show that 1/t ₀ ^t f(B _s ) ds converges to a constant z with an exponential rate in probability if and only if f has a uniform mean z. A similar result is also established in the case of random walks.     

A new class of symmetric polynomials defined in terms of tableaux

A new class of symmetric polynomials in n variables z = (z₁,…, z_n), denoted t_λ(z), and labelled by partitions λ = [λ₁ … λ_n] is defined in terms of standard tableaux (equivalently, in terms of Gel'fand-Weyl patterns of the general linear group GL(n,C)). The t_λ(z) are shown to be a -basis of the ring of all symmetric polynomials in n variables. In contrast to the usual basis sets such as the Schur functions e_λ(z), which are homogeneous polynomials in the z_i, the t_λ(z) are inhomogeneous. This property is reflected in the fact that the t_λ(z) are a natural basis for the expansion of certain (inhomogeneous) symmetric polynomials constructed from rising factorials. This and several other properties of the t_λ(z) are proved. Two generalizations of the t_λ(z) are also given. The first generalizes the t_λ(z) to a 1-parameter family of symmetric polynomials, T_λ(α; z), where α is an arbitrary parameter. The T_λ(α; z) are shown to possess properties similar to those of the t_λ(z). The second generalizes the t_λ(z) to a class of skew-tableau symmetric polynomials, t_λ/μ(z), for which only a few preliminary results are given.     

The Operator Factorization Method in Inverse Obstacle Scattering

The standard factorization method from inverse scattering theory allows to reconstruct an obstacle pointwise from the normal far field operator F. The kernel of this method is the study of the first kind Fredholm integral equation (F^* F)^1/4 f = Φ_z with the right-hand part In this paper we extend the factorization method to cover some kinds of boundary conditions which leads to non-normal far field operators. We visualize the scatterer explicitly in terms of the singular system of the selfadjoint positive operator F_# = [(ReF)^* (ReF)]^1/2 + ImF. The following characterization criterium holds: a given point z is inside the obstacle if and only if the function Φ_z belongs to the range of F_#^1/2. Our operator approach provides the tool for treatment of a wide class of inverse elliptic problems.