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Theory of Probability and its Applications

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2012

Volume 56, Issue 1, pp. 1-179


Approximating the Inverse of Banded Matrices by Banded Matrices with Applications to Probability and Statistics

P. Bickel and M. Lindner

Theory Probab. Appl. 56, pp. 1-20 (20 pages)

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In the first part of this paper we give an elementary proof of the fact that if an infinite matrix $A$, which is invertible as a bounded operator on $\ell^2$, can be uniformly approximated by banded matrices, then so can the inverse of $A$. We give explicit formulas for the banded approximations of $A^{-1}$ as well as bounds on their accuracy and speed of convergence in terms of their bandwidth. We then use these results to prove that the so-called Wiener algebra is inverse closed. In the second part of the paper we apply these results to covariance matrices $\Sigma$ of Gaussian processes and study mixing and beta mixing of processes in terms of properties of $\Sigma$. Finally, we note some applications of our results to statistics.

Chebyshev-Type Exponential Inequalities for Sums of Random Vectors and for Trajectories of Random Walks

A. A. Borovkov and A. A. Mogulskii

Theory Probab. Appl. 56, pp. 21-43 (23 pages)

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We obtain analogues of the well-known Chebyshev's exponential inequality ${\bf P}(\xi \ge x)\le e^{-\Lambda^{(\xi)}(x)}$, $x>{\bf E}\,\xi,$ for the distribution of a random variable $\xi$, where $\Lambda^{(\xi)}(x):=\sup_\lambda\{\lambda x- \log {\bf E}\,e^{\lambda \xi}\}$ is the large deviation rate function for $\xi$. Generalizations of this relation are established for multivariate random vectors $\xi$, for sums of the vectors, and for trajectories of random processes associated with such sums.

Uncertain Change-Point Problem for Stochastic Sequence

B. S. Darkhovsky

Theory Probab. Appl. 56, pp. 44-56 (13 pages)

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The change-point problem for a sequence of independent random variables is considered. Distributions of random variables before and after a change-point are unknown, but a finite collection of possible distributions is known a priori. Therefore, the problem is to detect the change without any information about its direction (an “uncertain change-point problem”). A new vector criterion to be minimized is proposed for change-point detection method quality estimation. For this criterion, nonasymptotic lower bounds are obtained. A method of quickest detection of the uncertain change-point is proposed for which these lower bounds are asymptotically attained.

Trace Approximations of Products of Truncated Toeplitz Operators

M. S. Ginovyan and A. A. Sahakyan

Theory Probab. Appl. 56, pp. 57-71 (15 pages)

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The paper establishes error orders for integral limit approximations to the traces of products of truncated Toeplitz operators generated by integrable real symmetric functions defined on the real line. These approximations and the corresponding error bounds are of importance in the statistical analysis of continuous-time stationary processes (asymptotic distributions and large deviations of Toeplitz-type quadratic functionals, estimation of the spectral parameters and functionals, etc.). An explicit second-order asymptotic expansion is found for the trace of a product of two truncated Toeplitz operators generated by the spectral densities of continuous-time stationary fractional Riesz–Bessel motions. The order of magnitude of the second term in this expansion is shown to depend on the long-memory parameters of the processes. Also, it is shown that the pole in the first-order approximation is removed by the second-order term, which provides a substantially improved approximation to the original functional.

Recurrence Relations for Price Bounds of Contingent Claims in Discrete Time Market Models

D. B. Rokhlin

Theory Probab. Appl. 56, pp. 72-95 (24 pages)

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We reduce the computation of price bounds of contingent claims to a sequence of interconnected finite dimensional optimization problems, depending on a parameter. In a perfect market model the obtained formulas characterize upper hedging price, while in a multicurrency market model with transaction costs they describe the set of initial portfolios, allowing for superhedging of a vector contingent claim. The mentioned formulas do not contain martingale measures or their analogues. The proofs are based on the martingale selection theorem. The effectiveness of the proposed approach is illustrated by several examples.

On Uniqueness of a Probability Solution to the Cauchy Problem for the Fokker–Planck–Kolmogorov Equation

S. V. Shaposhnikov

Theory Probab. Appl. 56, pp. 96-115 (20 pages)

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We obtain sufficient conditions for the uniqueness of a solution to the Cauchy problem on the whole space for the Fokker–Planck–Kolmogorov equation for probability measures under broad assumptions on coefficients and for arbitrary initial conditions. In particular, we do not assume any restrictions on the coefficients growth.

In memory of Anatolii Vladimirovich Skorokhod (1930–2011)

I. A. Ibragimov

Theory Probab. Appl. 56, pp. 116-119 (4 pages)

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A review of the career of Anatolii Vladimirovich Skorokhod, who passed away on January 4, 2011. He was an outstanding scientist, the author of fundamental works in the theory of probability and statistics, and academician of the National Ukrainian Academy of Sciences.

Ergodicity of a Multichannel Queueing System with Balking

T. N. Belorusov

Theory Probab. Appl. 56, pp. 120-126 (7 pages)

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In this paper we establish an ergodicity condition for a $G|G|r$ queueing system with balking. The input is considered to be regenerative. The proof is based on constructing a majorizing system in which probabilities of joining the system take two values.

The Distribution of the Number of Crossings of a Strip by Paths of the Simplest Random Walks and of a Wiener Process with Drift

I. S. Borisov and N. N. Nikitina

Theory Probab. Appl. 56, pp. 126-132 (7 pages)

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We compute the distribution of the number of crossings of a canonical strip by paths of a homogeneous random walk with a three-point distribution of the jump, defined on a finite or an infinite time-interval. By the Donsker–Prokhorov invariance principle (for an array of the random walks), we derive a similar formula for a Wiener process with drift from the above-mentioned result.

On Parameter-Measurability of the Stochastic Integral with Respect to the Two-Parameter Strong Martingale

N. A. Kolodii

Theory Probab. Appl. 56, pp. 132-140 (9 pages)

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This paper contains sufficient conditions of measurability with respect to a parameter of the limit of a sequence of random fields with trajectories in ${\mathbb D}$. Measurability with respect to a parameter of the quadratic variation of the two-parameter strong martingale was proved. We obtain sufficient conditions of measurability with respect to a parameter of the stochastic integral with respect to the two-parameter strong martingale.

On the Distribution of Time Spent by a Markov Chain at Different Levels Until Achieving a Fixed State

Ya. A. Lyulko

Theory Probab. Appl. 56, pp. 140-149 (10 pages)

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In this paper we consider the question of finding a distribution of the time spent by homogeneous Markov chain $Z=(Z_k)_{k\ge 0}$ (with countable state space $E$) at different levels of state space until first reaching a fixed point $b\in E.$ The work consists of two parts. In the first part we show that in the general case the distribution of residence time is geometric (with weight in zero). As an example we consider a skew random walk $S^{\alpha}=(S^{\alpha}_k)_{k\ge 0}$ with parameter $\alpha\in [0,\, 1].$ In this case we obtain the distribution in explicit form. In the second part of the paper we pass to the weak limit from residence time of skew random walk to the local time of skew Brownian motion $W^{\alpha}=(W^{\alpha}_t)_{t\ge 0}$ by using the extended Donsker–Prokhorov invariance principle established in [A. S. Cherny, A. N. Shiryaev, and M. Yor, Theory Probab. Appl., 47 (2003), pp. 377–394].

On $\bR^+$-Weakly Stable Distribution

G. Mazurkiewicz

Theory Probab. Appl. 56, pp. 149-154 (6 pages)

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A random vector ${\bf X}$ is called ${\bf R}^+$-weakly stable if for all (nonnegative) random variables $\Theta_1$ and $\Theta_2$ independent of ${\bf X}$, ${\bf X}^\prime$ there exists a (nonnegative) random variable $\Theta$ independent of ${\bf X}$ such that $\Theta_1{\bf X}+\Theta_2{\bf X}^\prime \stackrel{d}{=} \Theta{\bf X}.$ In this paper, as an answer to the open question given in [J. K. Misiewicz K. Oleszkiewicz and K. Urbanik, Studia Math., 167 (2005), pp. 195–213] we show that ${\bf R}^+$-weakly stable distributions have the same properties and stochastic structure as weakly stable distributions.

On Limit Distribution of Maximal Deviation of Empirical Distribution Density and Regression Function. II

M. S. Muminov

Theory Probab. Appl. 56, pp. 155-166 (12 pages)

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In this paper, for an unknown distribution function $f(t)$, $t\in {\bf R}^\nu$, a random vector $X\in {\bf R}^{\nu}$, and a regression function $r(t)={\bfE}\,(Y\,|\,X=t)$ of a random vector $(X,Y)$, $X\in {\bf R}^{\nu}$, $Y\in {\bf R}^{1}$, nonparametric kernel estimates $f_n(t)$ and $r_n(t)$ are constructed. It is proved that distribution of the maximal deviation of these estimators from the true distribution density $f(t)$ and the regression function $r(t)$ tend to the double exponential law as ${n \rightarrow \infty}$. With the aid of the constructed estimators we find a confidence region for $f(t)$ and $r(t)$, corresponding to the given confidence coefficient $\alpha$ $(0<\alpha <1)$, and construct a criterion for testing the hypothesis $H_0: f(t)=f_0(t)$ (respectively, $H_0': r(t)=r_0(t))$, where $f_0(t)$ is a given a priori distribution density, and $r_0(t)$ is a given function.

The Renewal Theorem in the Absence of Power Moments

S. V. Nagaev

Theory Probab. Appl. 56, pp. 166-175 (10 pages)

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The renewal theorem is proved for a random walk associated with a sequence of independent identically distributed random variables with distributions slowly varying at infinity. This case was not previously considered in the renewal theory.

News of Scientific Life On the 80th Birthday of A. A. Yushkevich

S. A. Molchanov

Theory Probab. Appl. 56, pp. 176-177 (2 pages)

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An article recognizing the 80th birthday of A. A. Yushkevich, with a review of his life and career.

Letters to the Editors

M. G. Shur

Theory Probab. Appl. 56, pp. 178-179 (2 pages)

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Errata to two papers which appeared in volume 55 of Theory of Probability and Its Applications.
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