Showing papers in "Theory of Probability and Its Applications in 2013"
TL;DR: A new form and a short complete proof of explicit two-sided estimates for the distribution function $F n,p}(k)$ of the binomial law with parameters $n,p$ from as mentioned in this paper were presented.
Abstract: We present a new form and a short complete proof of explicit two-sided estimates for the distribution function $F_{n,p}(k)$ of the binomial law with parameters $n,p$ from [D. Alfers and H. Dinges, Z. Wahrsch. Verw. Geb., 65 (1984), pp. 399--420]. These inequalities are universal (valid for all values of parameters and argument) and exact (namely, the upper bound for $F_{n,p}(k)$ is the lower bound for $F_{n,p}(k+1)$). Such estimates allow to bound any quantile of the binomial law by two subsequent integers that it contains.
55 citations
TL;DR: An upper bound on the risk of the selected estimator is derived and this result can be used in order to develop minimax and adaptive minimax estimators in specific nonparametric estimation problems.
Abstract: In the framework of an abstract statistical model we propose a procedure for selecting an estimator from a given family of linear estimators. We derive an upper bound on the risk of the selected estimator and demonstrate how this result can be used in order to develop minimax and adaptive minimax estimators in specific nonparametric estimation problems.
33 citations
TL;DR: In this article, Sobolev a priori estimates for the optimal transportation $T =
abla \Phi$ between probability measures $\mu=e^{-V} \, dx$ and $
u=e''-W}, dx$ on ${\bf R}^d'' were studied.
Abstract: We study Sobolev a priori estimates for the optimal transportation $T =
abla \Phi$ between probability measures $\mu=e^{-V} \, dx$ and $
u=e^{-W} \, dx$ on ${\bf R}^d$. Assuming uniform convexity of the potential $W$ we show that $\int \| D^2 \Phi\|^2_{HS} \, d\mu$, where $\|\cdot\|_{HS}$ is the Hilbert--Schmidt norm, is controlled by the Fisher information of $\mu$. In addition, we prove a similar estimate for the $L^p(\mu)$-norms of $\|D^2 \Phi\|$ and obtain some $L^p$-generalizations of the well-known Caffarelli contraction theorem. We establish a connection between our results and the Talagrand transportation inequality. We also prove a corresponding dimension-free version for the relative Fisher information with respect to the Gaussian measure.
24 citations
TL;DR: In this paper, it was shown that the constant square root of the moment generating function is the best possible condition for a multivariate distribution to be M-determinate, which is weaker than Cramer's condition requiring the existence of a moment generator.
Abstract: In probabilistic terms Hardy's condition is written as follows: ${\bf E}\,[e^{c{\sqrt X}}] 0$ a constant. If this holds, then all moments of $X$ are finite and the distribution of $X$ is uniquely determined by the moments (M-determinate). This condition, based on two papers by Hardy (1917/1918), is weaker than Cramer's condition requiring the existence of a moment generating function of $X$. We elaborate on Hardy's condition and show that the constant $1/2$ (square root) is the best possible for $X$ to be M-determinate. We describe relationships between Hardy's condition and properties of the moments of $X.$ We use this new condition to establish a result on the moment determinacy of an arbitrary multivariate distribution.
24 citations
TL;DR: In this article, a general Bayesian disorder detection problem for Brownian motion on a finite time segment is formulated, and properties of basic statistics are studied to reduce problems of quickest detection of disorder moments to optimal stopping problems.
Abstract: We formulate a general Bayesian disorder detection problem, which generalizes models considered in the literature. We study properties of basic statistics, which allow us to reduce problems of quickest detection of disorder moments to optimal stopping problems. Using general results, we consider in detail a disorder problem for Brownian motion on a finite time segment.
23 citations
TL;DR: In this paper, negative binomial approximation to sums of independent Z +$-valued random variables using Stein's method is employed to obtain the error bounds Convolution of negative Binomial and Poisson distribution is used as a three-parametric approximation.
Abstract: This paper deals with negative binomial approximation to sums of independent ${\bf Z}_+$-valued random variables Stein's method is employed to obtain the error bounds Convolution of negative binomial and Poisson distribution is used as a three-parametric approximation
22 citations
TL;DR: In this paper, the authors considered nonstationary continuous-time Markov chains for Markovian queues with bulk arrivals and group services and obtained the sharp bounds on the rate of convergence to the limiting regime, under the assumption that arrival and service rates do not depend on the length of the queue.
Abstract: We deal with nonstationary continuous-time Markov chains for Markovian queues with bulk arrivals and group services. Under the assumption that arrival and service rates do not depend on the length of the queue, we suggest the approach of obtaining the sharp bounds on the rate of convergence to the limiting regime. We also study the approximations of the limiting characteristics of the queue-length process.
19 citations
TL;DR: In this paper, it was shown that the absolute constant in nonuniform convergence rate estimates in the central limit theorem does not exceed $18.2 for sums of independent identically distributed random variables possessing absolute moments of the order 2 + ε with some ε > 0.
Abstract: We sharpen the upper bounds for the absolute constant in nonuniform convergence rate estimates in the central limit theorem for sums of independent identically distributed random variables possessing absolute moments of the order $2+\delta$ with some $0<\delta\le1$. In particular, it is demonstrated that under the existence of the third moment this constant does not exceed $18.2$. Also it is shown that the absolute constant in the estimates under consideration can be replaced by a function $C^*(|x|,\delta)$ of the argument $x$ of the difference of the prelimit and limit normal distribution functions for which a positive bounded nonincreasing majorant is found. Moreover, for $\delta=1$ this majorant is asymptotically exact (unimprovable) as $x\to\infty$ and sharpens the estimates due to Nikulin [preprint, arXiv:1004.0552v1 [math.ST], 2010] for all $x$. For the first time a similar result is obtained for the case $\delta\in(0,1)$. As a corollary, we obtain upper estimates for the Kolmogorov functions which ...
17 citations
TL;DR: The canonical genetic algorithm, which uses in its dynamics two parameters, namely mutation and crossover probabilities, which are kept fixed throughout the algorithm's evolution, will be allowed and the convergence of this new algorithm will be analyzed.
Abstract: Evolutionary algorithms are used to search for optimal points of functions. One of these algorithms, the canonical genetic algorithm, uses in its dynamics two parameters, namely mutation and crossover probabilities, which are kept fixed throughout the algorithm's evolution. In this paper, changes in those parameters will be allowed and the convergence of this new algorithm will be analyzed. We also present an approach to weak ergodicity of a nonhomogeneous Markov chains without using directly Dobrushin's $\delta$ coefficient.
15 citations
TL;DR: In this paper, exact small deviation asymptotics with respect to a weighted Hilbert norm for some well-known Gaussian processes were obtained without knowledge of the covariance operator of a weighted process.
Abstract: We find exact small deviation asymptotics with respect to a weighted Hilbert norm for some well-known Gaussian processes. Our approach does not require knowledge of the eigenfunctions of the covariance operator of a weighted process. Such a peculiarity of the method makes it possible to generalize many previous results in this area. We also obtain new relations connected to exact small deviation asymptotics for a Brownian excursion, a Brownian meander, and Bessel processes and bridges.
14 citations
TL;DR: In this article, the stability property in Cramer's characterization of the normal law is considered in the case of identically distributed summands, and instability is shown with respect to strong distances including the entropic distance to normality.
Abstract: The stability property in Cramer's characterization of the normal law is considered in the case of identically distributed summands. As opposite results, instability is shown with respect to strong distances including the entropic distance to normality (addressing a question of M. Kac).
TL;DR: Zitlukhin and Muravlev as mentioned in this paper considered Chernoff's problem of sequential testing of two hypotheses about the sign of the drift of a Brownian motion under the assumption that it is normally distributed.
Abstract: This paper contains detailed exposition of the results presented in the short communication [M. V. Zhitlukhin and A. A. Muravlev, Russian Math. Surveys, 66 (2011), pp. 1012--1013]. We consider Chernoff's problem of sequential testing of two hypotheses about the sign of the drift of a Brownian motion under the assumption that it is normally distributed. We obtain an integral equation which characterizes the optimal decision rule and find its solution numerically.
TL;DR: In this paper, it was shown that a fractional Brownian motion can be decomposed as an independent sum of a bifractional Brownians motion and a trifractionally Brownians' motion.
Abstract: Starting with a discussion about the relationship between the fractional Brownian motion and the bifractional Brownian motion on the real line, we find that a fractional Brownian motion can be decomposed as an independent sum of a bifractional Brownian motion and a trifractional Brownian motion that is defined in the paper. More generally, this type of orthogonal decomposition holds for a large class of Gaussian or elliptically contoured random functions whose covariance functions are Schoenberg--Levy kernels on a temporal, spatial, or spatio-temporal domain. Also, many self-similar, nonstationary (Gaussian, elliptically contoured) random functions are formulated, and properties of the trifractional Brownian motion are studied. In particular, a bifractional Brownian motion in $\bR^d$ is shown to be a quasi-helix in the sense of Kahane.
TL;DR: In this article, the authors introduced the notion of a continuous disintegration, which is a regular conditional probability measure which varies continuously in the conditioned parameter, and showed that for a certain quantity $M$ based on the covariance structure, $M < \oo$ is a necessary and sufficient condition for a Gaussian measure to have a continuous decomposition.
Abstract: The goal of this paper is to understand the conditional law of a stochastic process once it has been observed over an interval. To make this precise, we introduce the notion of a continuous disintegration: a regular conditional probability measure which varies continuously in the conditioned parameter. The conditioning is infinite dimensional in character, which leads us to consider the general case of probability measures in Banach spaces. Our main result is that for a certain quantity $M$ based on the covariance structure, $M < \oo$ is a necessary and sufficient condition for a Gaussian measure to have a continuous disintegration. The condition $M < \oo$ is quite reasonable: for the familiar case of stationary processes, $M = 1$.
TL;DR: This work provides a unified proof of all similar problems for Brownian motion considered in the existing literature an “bang-bang''-type optimal solution to the optimal selling time of a stock.
Abstract: In recent years, there has been a number of works on finding the optimal selling time of a stock so that the expected ratio of its selling price to a certain benchmark (e.g., its ultimate highest price) over a finite time horizon is maximized. Although formulated in different settings, they all result in a “bang-bang''-type optimal solution, as was originally discovered by Shiryaev, Xu, and Zhou [Quant. Finance, 8 (2008), pp. 765--776], which can literally be interpreted as “buy-and-hold” or “sell-at-once” depending on the quality of the stock. In this paper, we first provide three algebraic conditions on a class of benchmarks and call any benchmark satisfying the three conditions an ${\cal R}$-invariant performance benchmark. We show that if $F$ is an ${\cal R}$-invariant performance benchmark, then the corresponding optimal stopping problem has a “bang-bang''-type optimal solution. Our work here provides a unified proof of all similar problems for Brownian motion considered in the existing literature an...
TL;DR: In this article, the authors studied the behavior of the concentration functions of the weighted sums of random matrices with respect to the arithmetic structure of coefficients, and proved refinements of recent results of Friedland and Sodin and Rudelson and Vershynin.
Abstract: Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. This paper deals with the question of the behavior of the concentration functions of the weighted sums $\sum_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure of coefficients $a_k$. The growing interest in this topic is related to studies of eigenvalues of random matrices. In this paper we formulate and prove some refinements of recent results of Friedland and Sodin and Rudelson and Vershynin.
TL;DR: In this article, the level sets of a stationary and isotropic Gaussian random field with smooth realizations are studied and the Hilbert space functional central limit theorem for their Hausdorff measure is established.
Abstract: The level sets of a stationary and isotropic Gaussian random field with smooth realizations are studied. We establish the Hilbert space functional central limit theorem for their Hausdorff measure.
TL;DR: For the uniform distance between the distribution function of the standard normal law and the distribution functions of the normalized sum of independent random variables with symmetric distribution functions, this paper showed that for all ε ≥ 0.
Abstract: For the uniform distance $\Delta_n$ between the distribution function of the standard normal law and the distribution function of the normalized sum of $n$ independent random variables $X_1,\ldots,X_n$ with symmetric distribution functions $F_1,\ldots,F_n$ and ${\mathbf E}\,|X_j|=\beta_{1,j}$, ${\mathbf E}\,X_j^2=\sigma_j^2$, ${j=1,\ldots,n}$, for all $n\ge1$ and $c\ge0$, the estimates $\Delta_n\le \frac{1/2+\varkappa+c}{\sqrt{2\pi}}\,\ell_n + \frac{1/2-\varkappa-c}{\sqrt{2\pi}B_n^3} \sum_{j=1}^n\beta_{1,j}\,\sigma_j^2 +\left\{ \begin{array}{ll} 4\ell_n^{7/6}\wedge A(c)\ell_n^{4/3}&\!\!\! \mbox{in the non-i.i.d.\ case,}\\[4pt] 2\ell_n^{3/2}\wedge A(c)\ell_n^2, &\!\!\! \mbox{in the i.i.d.\ case} \end{array}\right. \!$ are proved, where $B_n^2\!=\!\sum_{j=1}^n\sigma_j^2,$ $\ell_n\!=\!B_n^{-3}\sum_{j=1}^n\,\!\!{\mathbf E}\,|X_j|^3\!,$ $\varkappa=\sup_{x>0}(\cos x-1+x^2/2)/x^3=0.0991\ldots,$ the function $A(c)$ is unbounded as $c\to0$ and decreases monotonically, takes finite values for all $c>0$, and is spec...
TL;DR: In this paper, the authors derived the asymptotic Cauchy hitting distribution for a hyperbolic Brownian motion on the Poincare half-plane and the classical Euclidean Poisson kernel.
Abstract: For a hyperbolic Brownian motion on the Poincare half-plane ${\mathbf H}^2$, starting from a point $z=(\eta, \alpha)$ inside a hyperbolic disc $U$ of radius $\bar{\eta}$, we obtain the probability of hitting the boundary $\partial U$ at the point $(\bar \eta,\bar \alpha)$. For $\bar{\eta} \to \infty$ we derive the asymptotic Cauchy hitting distribution on $\partial{\mathbf H}^2$. In particular, it follows that the hyperbolic Brownian motion starting at $(x,y) \in {\mathbf H}^2$ “hits” the boundary of ${\mathbf H}^2$ at a point which is Cauchy distributed with scale parameter $y^{\prime}=y/(x^2+y^2)$ and position parameter $x^{\prime}=x/(x^2+y^2)$. For small values of $\eta$ and $\bar \eta$ we obtain the classical Euclidean Poisson kernel. The exit probabilities from a hyperbolic annulus in ${\mathbf H}^2$ of radii $\eta_1$ and $\eta_2$ are derived and the transient behavior of hyperbolic Brownian motion is considered. Similar probabilities are calculated also for a Brownian motion on the surface of the th...
TL;DR: In this paper, the invariance principle for the logarithm of the stochastic process was proved for a branching process in a random environment, where it is assumed that the distribution of the random variable belongs (without centering) to the domain of attraction of the two-sided stable law.
Abstract: Let $(p_{i},q_{i}) $, $i\in {\bf Z}$, be a sequence of independent identically distributed pairs of random variables, where $p_{0}+q_{0}=1$ and, in addition, $p_{0}>0$ and $q_{0}>0 $ a.s. We consider a random walk in the random environment $(p_{i},q_{i}) $, $i\in {\bf Z}$. This means that in a fixed random environment a walking particle located at some moment $n$ at a state $i$ jumps at moment $n+1$ either to the state $(i+1)$ with probability $p_{i}$ or to the state $(i-1)$ with probability $q_{i}$. It is assumed that the distribution of the random variable $\log (q_{0}/p_{0})$ belongs (without centering) to the domain of attraction of the two-sided stable law with index $\alpha \in (0,2] $. Let $T_{n}$ be the first passage time of level $n$ by the mentioned random walk. We prove the invariance principle for the logarithm of the stochastic process $\{T_{\lfloor ns\rfloor},s\in [0,1] \}$ as $n\to \infty$. This result is based on the limit theorem for a branching process in a random environment which allow...
TL;DR: In this article, the authors considered queueing systems with an infinite number of servers and identical service times during a busy period and derived the limiting distribution and the ergodicity condition for the process that defines the number of customers in the system, and the distribution function of the system's busy period was also found.
Abstract: We consider queueing systems with an infinite number of servers and identical service times during a busy period. For the process that defines the number of customers in the system the limiting distribution and the ergodicity condition are obtained. The distribution function of the system's busy period is also found. The results are applied to the analysis of the formation of queues at uncontrolled intersections of roads. For the models proposed, the limiting characteristics are obtained and heavy traffic performance is studied.
TL;DR: In this paper, order statistics and empirical mathematical expectation (also called the estimate of mathematical expectation in the literature) are considered in the case of infinitely increasing random variables, and a relationship with the Van der Waals law of corresponding states is shown.
Abstract: The order statistics and empirical mathematical expectation (also called the estimate of mathematical expectation in the literature) are considered in the case of infinitely increasing random variables. The Kolmogorov concept, which he used in the theory of complexity, and the relationship with thermodynamics, which was pointed out already by Poincare, are considered. We compare the mathematical expectation (which is a generalization of the notion of arithmetical mean, and is generally equal to infinity for any increasing sequence of random variables) with the notion of temperature in thermodynamics while deploying a certain analogue of nonstandard analysis. It is shown that there is a relationship with the Van der Waals law of corresponding states. A number of applications of this concept in economics, in internet information networks, and self-teaching systems are also considered.
TL;DR: In this article, an analytic expression for the variance of ratio of integral functionals of fractional Brownian motion which arises as an asymptotic variance of Pitman estimators for a location parameter of independent identically distributed observations is provided.
Abstract: We provide an analytic expression for the variance of ratio of integral functionals of fractional Brownian motion which arises as an asymptotic variance of Pitman estimators for a location parameter of independent identically distributed observations. The expression is obtained in terms of derivatives of a logarithmic moment of the integral functional of limit likelihood ratio process (LLRP). In the particular case when the LLRP is a geometric Brownian motion, we show that the established expression leads to the known representation of the asymptotic variance of Pitman estimator in terms of Riemann zeta-function.
TL;DR: In this article, a strengthening of the Darmois-Skitovich-Ramachandran theorem on characterization of normal distribution by independence property of linear statistics is presented. But this theorem is not applicable to linear statistics.
Abstract: We prove one strengthening of the Darmois--Skitovich--Ramachandran theorem on characterization of normal distribution by independence property of linear statistics.
TL;DR: The proofs require a Szego theorem for generalized Toeplitz matrices which is analogous to a result of Kac, Murdoch, and Szege [J. Rational Mech. Anal., 2 (1953), pp. 767--800].
Abstract: In this paper, we show large deviations for random step functions of type $Z_n(t)=\frac{1}{n}\sum_{k=1}^{[nt]}X_k^2,$ where $\{X_k\}_k$ is a stationary Gaussian process. We deal with the associated random measures $
u_n=\frac{1}{n}\sum_{k=1}^nX_k^2 \delta_{k/n}$. The proofs require a Szego theorem for generalized Toeplitz matrices which is analogous to a result of Kac, Murdoch, and Szego [J. Rational Mech. Anal., 2 (1953), pp. 767--800]. We also study the polygonal line built on $Z_n(t)$ and show moderate deviations for both random families.
TL;DR: The asymptotic constant in the Berry-Esseen inequality for interval probabilities was shown to be O(2/π) in this paper, where σ 2/π is the number of intervals in the interval probability distribution.
Abstract: The asymptotic constant in the Berry--Esseen inequality for interval probabilities is shown to be $\sqrt{2/\pi}$.
TL;DR: In this paper, the authors consider linear Hamiltonian systems of arbitrary finite dimension and prove that, under the condition that one distinguished coordinate is subjected to dissipation and white noise, for almost any Hamiltonians and almost any initial conditions, there exists a unique limiting distribution.
Abstract: It is known that a linear Hamiltonian system has too many invariant measures; thus the problem of convergence to Gibbs measure makes no sense. We consider linear Hamiltonian systems of arbitrary finite dimension and prove that, under the condition that one distinguished coordinate is subjected to dissipation and white noise, for “almost any” Hamiltonians and “almost any” initial conditions, there exists a unique limiting distribution. Moreover, this distribution is Gibbsian with the temperature depending on the dissipation and on the variance of the white noise.
TL;DR: In this paper, asymptotic expansions for densities in the central limit theorem were obtained in terms of Chebyshev-Hermitian moments, and explicit expressions for principal parts of the Edgeworth-Cramer expansion were given in terms.
Abstract: We obtain asymptotic expansions for densities in the central limit theorem. Explicit expressions for principal parts of the Edgeworth--Cramer expansion are given in terms of Chebyshev--Hermitian moments.
TL;DR: In this article, the authors considered a two-type pure decomposable branching process in a random environment and studied the joint conditional distribution of the number of particles in the population at the moments when the environment is very unfavorable for particles of the first type.
Abstract: A two-type pure decomposable branching process in a random environment is considered. Each particle of this process may produce offspring of its own type only. Let $\exp \{X_{k}(i)\}$ be the mean number of children produced by a particle of type $i=1,2$ of generation $k$. Assuming that $X_{k}(2)=-X_{k}(1)$ with probability 1 and that the random walk $S_{n}(1)=X_{1}(1)+\cdots +X_{n}(1)$ specified by the random environment is oscillating, we study the joint conditional distribution of the number of particles in the population at the moments $nt,0
TL;DR: In this article, the authors considered supercritical branching processes with discrete time, where each particle has several random properties, and they considered the maxima of these properties in the population.
Abstract: We consider supercritical branching processes with discrete time, where each particle has several random properties. We are interested in the maxima of these properties in the population. Two cases are considered: the regular case (when the number of direct descendants has finite first and second moments and the joint distribution of the properties belongs to the domain of attraction of some multidimensional extremal law), and a more general case. Classes of nondegenerate limit laws for multivariate extremes under linear normalization are described. Emphasis is put on dependence structures described by copulas. For them, we obtain a functional equation and prove a theorem about continuation. A connection with max-semistable distribution is established.