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## Über dieses Buch

The role of Yuri Vasilyevich Prokhorov as a prominent mathematician and leading expert in the theory of probability is well known. Even early in his career he obtained substantial results on the validity of the strong law of large numbers and on the estimates (bounds) of the rates of convergence, some of which are the best possible. His findings on limit theorems in metric spaces and particularly functional limit theorems are of exceptional importance. Y.V. Prokhorov developed an original approach to the proof of functional limit theorems, based on the weak convergence of finite dimensional distributions and the condition of tightness of probability measures.

The present volume commemorates the 80th birthday of Yuri Vasilyevich Prokhorov. It includes scientific contributions written by his colleagues, friends and pupils, who would like to express their deep respect and sincerest admiration for him and his scientific work.​

## Inhaltsverzeichnis

### When Knowing Early Matters: Gossip, Percolation and Nash Equilibria

Abstract
Continually arriving information is communicated through a network of n agents, with the value of information to the j’th recipient being a decreasing function of jn, and communication costs paid by recipient. Regardless of details of network and communication costs, the social optimum policy is to communicate arbitrarily slowly. But selfish agent behavior leads to Nash equilibria which (in the n limit) may be efficient (Nash payoff = social optimum payoff) or wasteful (0 < Nash payoff < social optimum payoff) or totally wasteful (Nash payoff = 0). We study the cases of the complete network (constant communication costs between all agents), the grid with only nearest-neighbor communication, and the grid with communication cost a function of distance. The main technical tool is analysis of the associated first passage percolation process or SI epidemic (representing spread of one item of information) and in particular its “window width”, the time interval during which most agents learn the item. In this version (written in July 2007) many arguments are just outlined, not intended as complete rigorous proofs. One of the topics herein (first passage percolation on the N ×N torus with short and long range interactions; Sect. 6.2) has now been studied rigorously by Chatterjee and Durrett[4].
David J. Aldous

### A Mathematical Model of Investment Incentives

Abstract
An investment timing problem which takes into account both taxation (including tax exemptions) and financing by credit is considered. This problem is reduced to the optimal stopping of a two-dimensional diffusion process. We give the solution to the investment timing problem as a function of parameters of the model, in particular, of the tax holiday duration and interest rate for borrowing. We study the question whether the higher interest rate for borrowing can be compensated by tax holidays.

### The Shape of Asymptotic Dependence

Abstract
Multivariate risk analysis is concerned with extreme observations. If the underlying distribution has a unimodal density then both the decay rate of the tails and the asymptotic shape of the level sets of the density are of importance for the dependence structure of extreme observations. For heavy-tailed densities, the sample clouds converge in distribution to a Poisson point process with a homogeneous intensity. The asymptotic shape of the level sets of the density is the common shape of the level sets of the intensity. For light-tailed densities, the asymptotic shape of the level sets of the density is the limit shape of the sample clouds. This paper investigates how the shape changes as the rate of decrease of the tails is varied while the copula of the distribution is preserved. Four cases are treated: a change from light tails to light tails, from heavy to heavy, heavy to light and light to heavy tails.
Guus Balkema, Paul Embrechts, Natalia Nolde

### Limit Theorems for Functionals of Higher Order Differences of Brownian Semi-Stationary Processes

Abstract
We present some new asymptotic results for functionals of higher order differences of Brownian semi-stationary processes. In an earlier work [8] we have derived a similar asymptotic theory for first order differences. However, the central limit theorems were valid only for certain values of the smoothness parameter of a Brownian semi-stationary process, and the parameter values which appear in typical applications, e.g. in modeling turbulent flows in physics, were excluded. The main goal of the current paper is the derivation of the asymptotic theory for the whole range of the smoothness parameter by means of using second order differences. We present the law of large numbers for the multipower variation of the second order differences of Brownian semi-stationary processes and show the associated central limit theorem. Finally, we demonstrate some estimation methods for the smoothness parameter of a Brownian semi-stationary process as an application of our probabilistic results.
Ole E. Barndorff-Nielsen, José Manuel Corcuera, Mark Podolskij

### Retrieving Information from Subordination

Abstract
We recall some instances of the recovery problem of a signal process hidden in an observation process. Our main focus is then to show that if $$(X_{s},\,s\,\geq \,0)$$ is a right-continuous process, $$Y _{t} = \int \limits _{0}^{t}X_{s}\mathrm{d}s$$ its integral process and $$\tau = (\tau _{u},u \geq 0)$$ a subordinator, then the time-changed process $$(Y _{\tau _{u}},\,u\,\geq \,0)$$ allows to retrieve the information about $$(X_{\tau _{v}},\,v\,\geq \,0)$$ when τ is stable, but not when τ is a gamma subordinator. This question has been motivated by a striking identity in law involving the Bessel clock taken at an independent inverse Gaussian variable.
Jean Bertoin, Marc Yor

### Asymptotic Expansions for Distributions of Sums of Independent Random Vectors

Abstract
We consider the asymptotic behavior of the convolution $${P}^{{_\ast}n}(\sqrt{n}A)$$ of a k-dimensional probability distribution P(A) as $$n \rightarrow \infty$$ for A from the σ-algebra $${\mathfrak{M}}^{k}$$ of Borel subsets of Euclidian space R k or from its subclasses (often appearing in mathematical statistics). We will deal with two questions: construction of asymptotic expansions and estimating the remainder terms by using necessary and sufficient conditions. The most widely and deeply investigated cases are those where $${P}^{{_\ast}n}(\sqrt{n}A)$$ are approximated by the k-dimensional normal laws $${\Phi }^{{_\ast}n}(A\sqrt{n})$$ or by the accompanying ones $${\mathit{e}}^{n(P-E_{0})}$$. In this and other papers, estimating the remainder terms, we extensively use the method developed in the candidate thesis of Yu. V. Prokhorov (Limit theorems for sums of independent random variables. Candidate Thesis, Moscow, 1952) (adviser A. N. Kolmogorov) and there obtained necessary and sufficient conditions (see also Prokhorov (Dokl Akad Nauk SSSR 83(6):797–800 (1952) (in Russian); 105:645–647, 1955 (in Russian)).
Algimantas Bikelis

### An Extension of the Concept of Slowly Varying Function with Applications to Large Deviation Limit Theorems

Abstract
Karamata’s integral representation for slowly varying functions is extended to a broader class of the so-called ψ-locally constant functions, i.e. functions f(x) > 0 having the property that, for a given non-decreasing function ψ(x) and any fixed v, f(x + vψ(x)) ∕ f(x) → 1 as x. We consider applications of such functions to extending known theorems on large deviations of sums of random variables with regularly varying distribution tails.
Alexander A. Borovkov, Konstantin A. Borovkov

### Optimal and Asymptotically Optimal Control for Some Inventory Models

Abstract
A multi-supplier discrete-time inventory model is considered as illustration of problems arising in applied probability. Optimal and asymptotically optimal control is established for all values of parameters involved. The model stability is also investigated.
Ekaterina Bulinskaya

### Levy Preservation and Associated Properties for f -Divergence Minimal Equivalent Martingale Measures

Abstract
We study such important properties of f-divergence minimal martingale measure as Levy preservation property, scaling property, invariance in time property for exponential Levy models. We give some useful decomposition for f-divergence minimal martingale measures and we answer on the question which form should have f to ensure mentioned properties. We show that f is not necessarily common f-divergence. For common f-divergences, i.e. functions verifying $${f}^{{\prime\prime}}(x)\,=\,a{x}^{\gamma },\,a > 0,\,\gamma \in \mathbb{R}$$, we give necessary and sufficient conditions for existence of f-minimal martingale measure.
Suzanne Cawston, Lioudmila Vostrikova

### Non-standard Limit Theorems in Number Theory

Abstract
We prove a non-standard limit theorem for a sequence of random variables connected with the classical Möbius function. The so-called Dickman-De Bruijn distribution appears in the limit. We discuss some of its properties, and we provide a number of estimates for the error term in the limit theorem.
Francesco Cellarosi, Yakov G. Sinai

### Additive Functions and Gaussian Measures

Abstract
In this paper we examine infinite dimensional analogs of the measure theoretic variations of Cauchy’s classical functional equation for additive functions. In particular, we show that the a naïve generalization of the finite dimensional statement fails in infinite dimensions and show how it has to be altered to make it true. In the process, we develop various techniques which lead naturally to results about the structure of abstract Wiener spaces.
Linan Chen, Daniel W. Stroock

### Free Infinitely Divisible Approximations of n-Fold Free Convolutions

Abstract
Based on the method of subordinating functions we prove a free analog of error bounds in classical Probability Theory for the approximation of n-fold convolutions of probability measures by infinitely divisible distributions.

### Accurate Approximation of Correlation Coefficients by Short Edgeworth-Chebyshev Expansion and Its Statistical Applications

Abstract
In Christoph, Prokhorov and Ulyanov (Theory Probab Appl 40(2):250–260, 1996) we studied properties of high-dimensional Gaussian random vectors. Yuri Vasil’evich Prokhorov initiated these investigations. In the present paper we continue these investigations. Computable error bounds of order O(n − 3) or O(n − 2) for the approximations of sample correlation coefficients and the angle between high-dimensional Gaussian vectors by the standard normal law are obtained. We give some numerical results as well. Moreover, different types of Bartlett corrections are suggested.
Gerd Christoph, Vladimir V. Ulyanov, Yasunori Fujikoshi

### The Stein-Tikhomirov Method and Berry-Esseen Inequality for Sampling Sums from a Finite Population of Independent Random Variables

Abstract
We present a simplified version of the Stein-Tikhomirov method realized by defining a certain operator in class of twice differentiable characteristic functions. Using this method, we establish a criterion for the validity of a nonclassical central limit theorem in terms of characteristic functions, in obtaining of classical Berry-Esseen inequality for sampling sums from finite population of independent random variables.
Shakir K. Formanov, Tamara A. Formanova

### On One Inequality for Characteristic Functions

Abstract
This paper deals with an inequality for characteristic functions. This inequality (see (3) below) founds connection between “measure of almost normality” and characteristic functions. Also an analysis of accuracy in the local limit theorem and connection between the central limit and local limit theorem are given.
Nicko Gamkrelidze

### On the Nonlinear Filtering Equations for Superprocesses in Random Environment

Abstract
In the paper we define the Dawson-Watanabe type superprocesses in random environment as solutions to the related martingale problems. An environment is modelled by a finite state time homogeneous Markov process with the given transition probability intensity matrix. A system of nonlinear stochastic equations is derived for a posteriori probabilities. Reduced system of linear equations is also obtained.
Bronius Grigelionis

### Upper Bounds for Bernstein Basis Functions

Abstract
From Markov’s bounds for binomial coefficients (for which a short proof is given) upper bounds are derived for Bernstein basis functions of approximation operators and their maximum. Some related inequalities used in approximation theory and those for concentration functions are discussed.
Vijay Gupta, Tengiz Shervashidze

### On Distribution of Zeros of Random Polynomials in Complex Plane

Abstract
Let $$G_{n}(z) = \xi _{0} + \xi _{1}z + \cdots + \xi _{n}{z}^{n}$$ be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of G n (z) are uniformly distributed in [0, 2π] asymptotically as $$n\,\rightarrow \,\infty$$. We also prove that the condition $$\mathbf{E}\,\ln (1 + \vert \xi _{0}\vert )\,<\,\infty$$ is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference.
Ildar Ibragimov, Dmitry Zaporozhets

### Dependence and Interaction in Branching Processes

Abstract
Independence of reproducing individuals can be viewed as the very defining property of branching processes. It is crucial for the most famous results of the theory, the determination of the extinction probability and the dichotomy between extinction and exponential increase. In general processes, stabilisation of the age-distribution under growth follows, and indeed of the over-all population composition, and so do the many fine results of the area, like conditional stabilisation of the size of non-extinct subcritical processes. The last two decades have witnessed repeated attempts at treating branching processes with various kinds of dependence between individuals, ranging from local dependence between close relatives only to population size dependence. Of particular interest are very recent findings on processes that change from being supercritical to subcriticality at some threshold size, the carrying capacity of the habitat. We overview the development with an emphasis on these recent results.
Peter Jagers, Fima C. Klebaner

### Testing Functional Connection between Two Random Variables

Abstract
Two discrete random variables, ξ and η are considered. The goal is to decide whether $$\eta$$ is a function of ξ. A series of tests are performed, $$(\xi _{i},\eta _{i}),1\,\leq \,i\,\leq \,m$$, are independent experiences with the same distribution as $$(\xi ,\eta )$$. The hypothesis is declined if $$\xi _{i} = \xi _{j},\eta _{i}\neq \eta _{j}$$ holds for some $$i\neq j$$. A condition is given on the character of convergence of the joint distribution of ξ and η ensuring the rejection of the hypothesis with a given limiting probability p.
Gyula O. H. Katona

### The Symmetry Group of Gaussian States in $${L}^{2}({\mathbb{R}}^{n})$$

Abstract
This is a continuation of the expository article by Parthasarathy (Commun Stoch Anal 4:143–160, 2010) with some new remarks. Let S n denote the set of all Gaussian states in the complex Hilbert space $${L}^{2}({\mathbb{R}}^{n}),$$ K n the convex set of all momentum and position covariance matrices of order 2n in Gaussian states and let $$\mathcal{G}_{n}$$ be the group of all unitary operators in $${L}^{2}({\mathbb{R}}^{n})$$ conjugations by which leave S n invariant. Here we prove the following results. K n is a closed convex set for which a matrix S is an extreme point if and only if $$S = \frac{1} {2}{L}^{T}L$$ for some L in the symplectic group $$Sp(2n, \mathbb{R}).$$ Every element in K n is of the form $$\frac{1} {4}({L}^{T}L + {M}^{T}M)$$ for some L, M in $$Sp(2n, \mathbb{R}).$$ Every Gaussian state in $${L}^{2}({\mathbb{R}}^{n})$$ can be purified to a Gaussian state in $${L}^{2}({\mathbb{R}}^{2n}).$$ Any element U in the group $$\mathcal{G}_{n}$$ is of the form $$U = \lambda W(\boldsymbol \alpha )\varGamma (L)$$ where λ is a complex scalar of modulus unity, $$\boldsymbol \alpha \in {\mathbb{C}}^{n},$$ $$L \in Sp(2n, \mathbb{R}),$$ $$W(\boldsymbol \alpha )$$ is the Weyl operator corresponding to $$\boldsymbol \alpha$$ and $$\varGamma (L)$$ is a unitary operator which implements the Bogolioubov automorphism of the Lie algebra generated by the canonical momentum and position observables induced by the symplectic linear transformation L.
Kalyanapuram R. Parthasarathy

### Solution of the Optimal Stopping Problem for One-Dimensional Diffusion Based on a Modification of the Payoff Function

Abstract
A problem of optimal stopping for one-dimensional time-homogeneous regular diffusion with the infinite horizon is considered. The diffusion takes values in a finite or infinite interval ]a,b[. The points a and b may be either natural or absorbing or reflecting. The diffusion may have a partial reflection at a finite number of points. A discounting and a cost of observation are allowed. Both can depend on the state of the diffusion. The payoff function g(z) is bounded on any interval [c, d], where a < c < d < b, and twice differentiable with the exception of a finite (may be empty) set of points, where the functions g(z) and $${g}^{{^\prime}}(z)$$ may have a discontinuities of the first kind. Let L be an infinitesimal generator of diffusion which includes the terms corresponding to the discounting and the cost of observation. We assume that the set $$\{z : Lg(z) > 0\}$$ consists of a finite number of intervals. For such problem we propose a procedure of constructing the value function in a finite number of steps. The procedure is based on a fact that on intervals where Lg(z) > and in neighborhoods of points of partial reflections, points of discontinuities, and points a or b in case of reflection, one can modify the payoff function preserving the value function. Many examples are considered.
Ernst Presman

### The Times of Yuri Vasilyevich Prokhorov

Abstract
Yuri Vasilyevich Prokhorov is the eminence grise of Russian probability theory. Every one of us interested in probability or asymptotic statistics has come across his celebrated weak compactness theorem at one time or another. He was interviewed earlier by Larry Shepp (Stat Sci 7: 123–130, 1992). That interview dealt largely with his impressive career and scientific work, his international contacts and the issue of discrimination in the Soviet Union. The world has changed considerably in the intervening years and our knowledge and perspective of the past has developed accordingly. It seemed natural to us to talk once more to the man who lived through these turbulent times as the intellectual heir of Kolmogorov, and as one who was in a position to observe the inner workings of the powerful Soviet (later Russian) Academy of Sciences, the Steklov Mathematical Institute in Moscow, and the activities of his many colleagues throughout the country and elsewhere. This interview took place between November 13 and 28, 2006 at Bielefeld University. As it was more like a friendly three-way conversation than a formal interview, we did not identify the two interviewers, but merely indicated the person who asked a question as “interviewer”.
Friedrich Götze, Willem R. van Zwet

### A Conversation with Yuri Vasilyevich Prokhorov

Abstract
Yuri Vasilyevich Prokhorov was born on December 15, 1929. He graduated from Moscow University in 1949 and worked at the Mathematical Institute of the Academy of Sciences from 1952, and as a Professor on the faculty of Moscow University since 1957. He became a corresponding member of the Academy in 1966 and an Academician in 1972. He received the Lenin Prize in 1970. The basic directions of his research are the theory of probability. He developed asymptotic methods in the theory of probability. In the area of the classical limit theorems, he studied the conditions of applicability of the strong law of large numbers and the so-called local limit theorems for sums of independent random variables. He proposed new methods for studying limit theorems for random processes; these methods were based on studying the convergence of measures in function space. He applied these methods to establish the limiting transition from discrete processes to continuous ones. He found (in 1953 and 1956) necessary and sufficient conditions for weak convergence in function spaces. He has several papers on mathematical statistics, on queuing theory and also on the theory of stochastic control. This conversation took place at the Steklov Institute in early September 1990. It was taped in Russian and translated by Abram Kagan. The final version was edited by Ingram Olkin.
Larry Shepp
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