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Theory of Probability and its Applications

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1959

Volume 4, Issue 4, pp. 339-443


Asymptotic Analysis of the Distribution of the Maximum Deviation in the Bernoulli Scheme

V. S. Korolyuk (Kiev)

Theory Probab. Appl. 4, pp. 339-366 (28 pages) | Cited 6 times

Online Publication Date: July 17, 2006

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Methods are given for constructing asymptotic expansions for the maximum deviation distribution in the Bernoulli scheme and in its limiting case, the Poisson scheme. The paper contains a review of results on the asymptoticity of distributions of the maximum deviations between a theoretical and an empirical distribution function, and also between two empirical distribution functions.

On the Maximum Partial Sums of Sequences of Independent Random Variables

R. P. Pakshirajan

Theory Probab. Appl. 4, pp. 367-372 (6 pages)

Online Publication Date: July 17, 2006

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In the present work the results of K. L. Chung [2] concerning the maximum partial sums of sequences of independent random variables are obtained for a weaker condition. The method employed in the proof is analogous to the one used by Chung with the difference that, instead of Esseen’s approximations involving third moments, we use Berry’s approximations involving only second moments.

On the Estimation of Regression Coefficients of a Continuous Parameter Time Series with a Stationary Residual

Chiang Tse-Pei

Theory Probab. Appl. 4, pp. 373-390 (18 pages) | Cited 4 times

Online Publication Date: July 17, 2006

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Let $y(t) = x(t) + m(t)$ be a complex probability process with mean value\[ {\bf E}y(t) = m(t) = \sum\limits_{\nu = 1}^S {\gamma _\nu \varphi _\nu (t)} \](where $\varphi _\nu (t)$, $\nu = 1,2, \cdots s$, are given functions of $t$, and $\gamma _\nu $, $\nu = 1,2, \cdots s$ are unkownconstants) and with stationary residual $x(t)$:\[ E(x(u)\overline {x(v)} ) = r(u - v).\]It is required to estimate the vector $\gamma = \{ \gamma _1 ,\gamma _2 , \cdots ,\gamma _s \} $ on the basis of one realization of the process $y(t)$ on the finite segment $ - T \leqq t \leqq T$.
In the case being considered the estimate of the vector $\gamma $ by the method of least squares obtained by minimizing the quadratic form\[ \int_{ - T}^T {\left| {y(t) - \sum\limits_{\nu = 1}^S {c_\nu \varphi _\nu (t)} } \right|^2 dt} \] is equal to\[ {}_L C_T = \Phi ^{ - 1} \left( {\begin{array}{*{20}c} {\int_{ - T}^T {y(t)} \overline {\varphi _1 (t)} dt} \\ \vdots \\ {\int_{ - T}^T {y(t)} \overline {\varphi _s (t)} dt} \\ \end{array} } \right), \]where\[ \Phi = \left[ {\Phi _{\mu \nu } } \right]_{1 \leqq \mu ,\nu \leqq s} ,\quad \Phi _{\mu \nu } = \int_{ - T}^T {\overline {\varphi _\mu (t)} \varphi _\nu (t)dt} .\]This estimate is unbiased.
In Section 2 of this article it is shown that in our case there exists a best linear unbiased estimate ${}_0 C_T $ (i.e. a linear unbiased estimate with a least dispersion matrix).
In Section 3 the asymptotic behavior of dispersion matrices ${\bf E}({}_L C_T - \gamma )\overline {({}_L C_T - \gamma )'} $ and ${\bf E}({}_0 C_T - \gamma )\overline {({}_0 C_T - \gamma )'} $ (the prime signifies a Hermitean-conjugate matrix) is considered. If (i) the stationary process $x(t)$ is regular, (ii) the functions $\varphi _\nu (t)$ are trigonometric, (iii) the spectral density of the process $x(t)$ is continuous in certain separate points, then the main terms of the matrices ${\bf E}({}_L C_T - \gamma )\overline {({}_L C_T - \gamma )'} $ and ${\bf E}({}_0 C_T - \gamma )\overline {({}_0 C_T - \gamma )'} $ as $T \to \infty $ coincide with each other. In other words, under the above-mentioned conditions, the asymptotic efficiency of the estimate ${}_L C_T $ is proved in the class of linear unbiased estimates.

Waveguides with Random Inhomogeneities and Brownian Motion In the Lobachevsky Plane

M. E. Gertsenshtein and V. B. Vasil’ev

Theory Probab. Appl. 4, pp. 391-398 (8 pages) | Cited 21 times

Online Publication Date: July 17, 2006

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It has been shown that the probability density for the continuous random process of the resultant of independent values which are summed up according to the linear-fractional law satisfies the diffusion equation in the Lobachevsky plane. Green’s function of the diffusion equation, which apparently is a new distribution, has been found.

Limit Theorems for the Compositions of Distributions in the Lobachevsky Plane and Space

F. I. Karpelevicii and V. N. Tutubalin

Theory Probab. Appl. 4, pp. 399-402 (4 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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Some physical problems require the study of random variables (or measures) on the Lobachevsky plane (space) $L$.
A Borel measure $\mu (\Gamma )$ is symmetric if $\mu (h\Gamma ) = \mu (\Gamma )$ for any Borel measurable set $\Gamma $ and any rotation $h$ around the origin of coordinates $O$. The composition $\mu _1 * \mu _2 (\Gamma )$ of the symmetric measures $\mu _1 (\Gamma )$ and $\mu _2 (\Gamma )$ is defined by the equtaion \[\mu _1 * \mu _2 (\Gamma ) = \int_L {\left( {\theta _x^{ - 1} \Gamma } \right)} \mu _2 (dx)\] where $\theta _x $ denotes any motion in $L$ which transforms $O$ into $x,\mu _1 * \mu _2 (\Gamma )$ being equal to $\mu _2 * \mu _1 (\Gamma )$.
For symmetric measures the concept of a characteristic function is introduced. The characteristic functions of two measures are multiplied to get the characteristic function for the composition of this measures.
In the association of Borel probability measures there is a system of normal measures. If the set of symmetric measures $\mu _{n,r} ,n \geqq l,l \leqq r \leqq k_n $, satisfies some conditions analogous to the Lindeberg-Feller ones, the sequence of the measures\[ \xi _n = \mu _{n,1} * \cdots * \mu _{n,k_n } \]weakly converges to a normal measure.

Analytic Random Processes

Yu. K. Belyaev

Theory Probab. Appl. 4, pp. 402-409 (8 pages) | Cited 14 times

Online Publication Date: July 17, 2006

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This paper is devoted to investigating the so-called analytic random processes. Random process $\xi (t)$ is called analytic in a region $D$ if almost all its sample functions are analytic and possess an analytic continuation in the region $D$. Analyticity of the covariance function $B(t,s) = {\bf M}\xi (t)\xi (s)$ in the neighborhood of $(t_0 ,t_0 )$ is a sufficient condition for analyticity of $\xi (t)$ in the neighborhood of $t_0 $. For Gaussian processes, this condition is also necessary. Some other problems connected with analytic processes are also investigated.

Some Remarks on Goncharov’s Paper from the Domain of Combinatorics

V. I. Babkin, P. F. Belyaev, and Yu. I. Maksimov

Theory Probab. Appl. 4, pp. 409-414 (6 pages)

Online Publication Date: July 17, 2006

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This note contains some results on the asymptotic distribution of the random vector $(\nu _1 ,\nu _2 , \cdots ,\nu _{k - 1} ,\nu _k )$, where $\nu _1 ,\nu _2 , \cdots ,\nu _{k - 1} ,\nu _k $ are the numbers of $A$-series of lengths $1,2, \cdots ,k - 1$ greater or equal to $k$, respectively, in the simple homogeneous Markov chain with two states $A$ and $B$. The asymptotic distribution of the above-mentioned vector (when appropriately formed) is shown to be multivariate normal with the parameters of the distribution calculated.
Possible extensions for a number of states greater than two are also discussed.

On the Estimation of the Mean in Stationary Processes

S. Ya. Vilenkin

Theory Probab. Appl. 4, pp. 415-416 (2 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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The variance of the estimate\[ m_{N + 1} = \frac{1}{{N + 1}}\sum\limits_{i = 0}^N {\xi \left( {\frac{i}{N} \cdot T} \right)} \] of a mean of a stationary process ${\xi (t)}$ is shown to attain its minimum value for some finite $N$.

On the Unimodality of Geometric Stable Laws

I. A. Ibragimov and K. E. Chernin

Theory Probab. Appl. 4, pp. 417-419 (3 pages) | Cited 9 times

Online Publication Date: July 17, 2006

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It is proved that all distribution functions of stable laws are unimodal.

On One Theorem of R. Bellman

I. V. Romanovskii

Theory Probab. Appl. 4, pp. 420-421 (2 pages)

Online Publication Date: July 17, 2006

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This paper contains a proof of a theorem which R. Bellman [1], [2] proved incorrectly.

On the Theory of Optimal Filtering of a Signal in the Presence of Internal Random Noise

Li Heng Won

Theory Probab. Appl. 4, pp. 422-426 (5 pages)

Online Publication Date: July 17, 2006

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This paper is devoted to the solution of a synthesis problem in optimal automatic control systems with internal stationary random noise applied to different points in the system.

On the Extrapolation of Generalized Stationary Random Processes

Yu. A. Rozanov

Theory Probab. Appl. 4, pp. 426-431 (6 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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Let $D$ denote the space of all infinitely differentiable functions $\varphi $ that are equal to zero outside an interval; $\xi (\varphi )$ is a random distribution, $H$ is a linear closure of random variables, $\xi (\varphi ) \cdot \varphi \in D$; and $H_S^ - $ is a linear closure of random variables ($\xi (\varphi)$ (where $\varphi \in D$ and $\varphi (t) = 0$ for $t \geqq S$).
The random distribution $\xi (\varphi )$ is called singular if $H = \cap _S H_S^ - $ and regular if $ \cap _S = H_S^ - = 0$. The necessary and sufficient conditions for singularity (resp. regularity) of the random distribution $\xi (\varphi )$ are given.
The problem of extrapolation is solved for the case where $\xi (\varphi )$ is regular.

Summary of Papers Presented at The Sessions of the Probability Research Seminar, Moscow University (February–May 1959)

V. V. Petrov

Theory Probab. Appl. 4, pp. 432-443 (12 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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