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Theory of Probability and its Applications

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1964

Volume 9, Issue 4, pp. 521-687


Age-Dependent Branching Processes

B. A. Sevast’yanov

Theory Probab. Appl. 9, pp. 521-537 (17 pages) | Cited 5 times

Online Publication Date: July 28, 2006

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In § 1 we consider a generalization of one model of age-dependent branching processes constructed by R. Bellman and T. E. Harris. In § 2 we derive necessary and sufficient conditions for the extinction of such branching processes. In § 3, 4 we prove a limit theorem for critical branching processes.

Convergence of Conditional Expectation Operators

K. Krickeberg

Theory Probab. Appl. 9, pp. 538-549 (12 pages)

Online Publication Date: July 28, 2006

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Two concepts of convergence for nets of $\sigma $-rings are defined for every Orlicz space $\mathfrak{L}_M^ * $, which imply essential, respectively, stochastic convergence of the corresponding conditional expectations of any random variable in $\mathfrak{L}_M^ * $.

Some Limit Theorems in the Theory of Mass Service

A. A. Borovkov

Theory Probab. Appl. 9, pp. 550-565 (16 pages) | Cited 5 times

Online Publication Date: July 28, 2006

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In the paper limit theorems describing the work of a queuing system with one server are considered in the general case. In § 1 the probabilities of large values of the waiting time are studied. The absence of the explicit form of the distributions does not prevent here the detailed study of the asymptotic expressions for the required probabilities. § 2 contains the refinement of the limit theorems obtained earlier in [2]–[5] and describes the work of the system under the condition of heavy traffic.

Investigation of Conditions for the Asymptotic Existence of the Configuration Integral of Gibbs’ Distribution

R. L. Dobrushin

Theory Probab. Appl. 9, pp. 566-581 (16 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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Let $V$ be a cube of dimension $\nu $, with volume $|V|$. Let ${{|V|} / {N \to \nu }}$, $N \to \infty $. Let ${\bf x} = (x_1 , \cdots ,x_N )$, $x_i \in V$, $i = 1, \cdots ,N$,\[ Q(V,N) = \int_V { \cdots \int_V {\exp \{ { - \beta U({\bf x})} \}} } dx_1 \cdots dx_N , \] where \[ U({\bf x}) = \sum\limits_{1 \leqq i < j \leqq N} {\Phi \left( {\left| {x_i - x_j } \right|} \right)} . \] The conditions on $\Phi (y)$, which are sufficient and in some sense necessary for the existence of the finite limit \[ \mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\log \frac{1}{{N!}}Q(V,N) \] are given.

Boundary Conditions for Certain Markov Processes

A. V. Skorokhod

Theory Probab. Appl. 9, pp. 582-590 (9 pages)

Online Publication Date: July 28, 2006

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In the article a boundary for a Markov process is defined. This boundary is connected with the harmonic functions of the process and is analogous to the Martin boundary. It is established that all the functions belonging to the domain of the infinitesimal operator of the process satisfy a certain condition on this boundary. This condition and the characteristic operator of the process determine the process completely.

On the Control of Non-Terminating Diffusion Processes

Petr Mandl

Theory Probab. Appl. 9, pp. 591-603 (13 pages) | Cited 4 times

Online Publication Date: July 28, 2006

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In Part I of the paper the mean cost for a unit of time arising from a non-terminating diffusion process, denoted by $\Theta $, is defined. One part of the cost originates from the motion inside the interval between two boundaries, the other part originates in the jumps from these boundaries. $\Theta $ is characterised by Theorem I. In Part II it is supposed that the diffusion coefficient and the coefficient of the local shift of the process depend on a control variable. The optimum $\hat\Theta $ of realizable mean costs may be determined by means of Theorem 2.

On Markov Sufficient Statistics in Non-Additive Bayes Problems of Sequential Analysis

A. N. Shiryaev

Theory Probab. Appl. 9, pp. 604-618 (15 pages) | Cited 10 times

Online Publication Date: July 28, 2006

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The question of finding Markov sufficient statistics (see definition 3) in the problem of minimisation of the functional (2) is considered. It is supposed that the parameter $\theta $ is the random moment at which the density $f_0 $ changes to $f_1 $ (§ 1–3) or to one of the $f_1 , \cdots ,f_m $ (§ 5). In the case when the densities $f_0 ,f_1 , \cdots ,f_m $ belong to the exponential family and the functional which is minimized is a non-additive one of a special form, we find a finite number of Markov sufficient statistics. Connections between the problem considered and other problems of sequential analysis are also discussed.

Distributions Related to the Hypergeometric Distribution

L. N. Bol’shev

Theory Probab. Appl. 9, pp. 619-624 (6 pages) | Cited 2 times

Online Publication Date: July 28, 2006

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The concept of the simple sequential scheme [1] is extended to the case of samples without replacement from a finite universe. For these schemes the resulting probability distributions can be expressed by formula (8) in terms of the usual hypergeometric distribution.

A Characteristic Property of the Normal Distribution

A. A. Zinger and Yu. V. Linnik

Theory Probab. Appl. 9, pp. 624-626 (3 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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As it is well known the normal distribution is characterized by the uniformity of the distribution of the random vector $({{(X_1 - \bar X)} / {s, \cdots ,{{(X_n - \bar X)} / s}}})$ on the unit sphere (here we use usual notations). It is shown that there exists a set of triplets of points of that sphere such that the normality of the sample follows from the constancy of the density of that vector only on any one of these triplets.

On the Asymptotic Behavior of the Prediction Error

I. A. Ibragimov

Theory Probab. Appl. 9, pp. 627-634 (8 pages) | Cited 2 times

Online Publication Date: July 28, 2006

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Let $\{ {x_j } \}$ be a stationary stochastic process in the wide sense which is regular, with spectral density function $f(\lambda )$. Denote by $\sigma _n^2 $ the mean square prediction error in predicting $x_0 $ by linear forms in $x_{ - 1} ,x_{ - 2} , \cdots ,x_{ - n} $. Let $\delta _n = \sigma _n^2 - \sigma _\infty ^2 = \sigma _n^2 - \sigma ^2 $. The rate of convergence $\delta _n \downarrow 0$ is investigated in this article.

Search by an Oval

E. Gyachyauskas

Theory Probab. Appl. 9, pp. 634-637 (4 pages)

Online Publication Date: July 28, 2006

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Let $H$ be a circle of radius $R$ containing a target $K_0 $. The probability of the intersection of $K_0 $ with a randomly chosen oval $K$ is computed. Assuming $K_0 $ to be an oval and $K$ to be an ellipse of a fixed area $F$ we indicate parameters $a$ and $b$, for which $P$ attains its maximum.

On Statistical Quadratures

E. Gyachyauskas

Theory Probab. Appl. 9, pp. 637-640 (4 pages)

Online Publication Date: July 28, 2006

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Some statistical methods of estimation of the area of a plane domain are considered. They are compared with the methods exploited in the book by F. Chayes [3].

A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes

V. P. Chistyakov

Theory Probab. Appl. 9, pp. 640-648 (9 pages) | Cited 51 times

Online Publication Date: July 28, 2006

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Let $\xi _1 , \cdots ,\xi _n , \cdots $ be independent random positive variables and let ${\bf P} \{ {\xi _k < t} \} = G(t)$, $k = 1, \cdots ,n, \cdots $ Let us denote\[ {\bf P}\left\{ {\xi _1 + \cdots + \xi _n < t} \right\} = G_n (t). \]
Theorem.\[ \mathop {\lim }\limits_{t \to \infty } \frac{{1 - G_n (t)}}{{1 - G(t)}} = n,\qquad n = 1,2,3, \cdots , \]if and only if\[ \mathop {\lim }\limits_{t \to \infty } \frac{{1 - G_2 (t)}}{{1 - G(t)}} = 2. \] This theorem is useful in some investigations of age-dependent branching processes.

On the Calculation of the Power of the Test of Empty Boxes

V. P. Chistyakov

Theory Probab. Appl. 9, pp. 648-653 (6 pages) | Cited 9 times

Online Publication Date: July 28, 2006

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Let us suppose that $n$ balls are distributed among $N$ boxes, so that each ball may fall into the i-th box with probability $a_i $, $a_i \geqq 0$, $\sum\nolimits_{i = 1}^N {a_i = 1} $, independently of what happens to the other balls. Let $\mu _0 $ denote the number of boxes which remain empty. In [5] the proof of the theorem on asymptotic normality of $\mu _0 $ under the assumption (1) is not correct. In the present paper a more general theorem on asymptotic normality of $\mu _0 $ is proved.

The First Passage Time of a Level and the Behavior at Infinity for a Class of Processes with Independent Increments

V. M. Zolotarev

Theory Probab. Appl. 9, pp. 653-662 (10 pages) | Cited 6 times

Online Publication Date: July 28, 2006

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Let $\mathfrak{S}$ be the class of homogeneous processes $\xi (t)$ with independent increments and without positive jumps, and let $\mathfrak{S}^ + $, $\mathfrak{S}^ - $ be its subclasses such that\[ \begin{gathered} {\bf P}\left\{ {\mathop {\sup }\limits_t \xi (t) = \infty } \right\} = 1,\qquad \xi (t) \in \mathfrak{S}^ + , \hfill \\ {\bf P}\left\{ {\mathop {\inf }\limits_t \xi (t) = - \infty } \right\} = 1,\qquad \xi (t) \in \mathfrak{S}^ - . \hfill \\ \end{gathered} \]
It is proved that $\xi \in \mathfrak{S}^ + $ (resp. $\xi \in \mathfrak{S}^ - $) if and only if $\gamma = {\bf E}\xi (1) \geqq 0$ (resp. $\gamma \leqq 0$). The exact expression for the distribution of the first intersection moment $\tau _x $ of a level $x > 0$ by a process $\xi \in \mathfrak{S}^ + $ is found. We have also found the exact expression for the probability $P(x) = {\bf P}\{ {\sup _t \xi (t) \geqq x} \}$ as well as the implicit dependence of $Q(x) = {\bf P}\{ {\inf _t \xi (t) \leqq - x} \}$ on the distribution of $\xi (t)$. Asymptotic and other properties of the probabilities $P$, $Q$ are investigated.

On a Local Limit Theorem for Lattice Random Variables

N. G. Gamkrelidze

Theory Probab. Appl. 9, pp. 662-664 (3 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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Yu. V. Prokhorov made a conjecture that the two statements are equivalent: (a) the local limit theorem is applicable to a sequence of integer-valued random variables, (b) these variables obey the central limit theorem and their sums are asymptotically uniformly distributed modulo $h$ for arbitrary $h$.
We give an example showing that the above conjecture does not hold true.

On Normal Distributions in $l_p $

N. N. Vakhaniya

Theory Probab. Appl. 9, pp. 665-665 (1 page)

Online Publication Date: July 28, 2006

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The note contains two theorems. The first theorem states that in $l_p $ a normal measure integrates any power of $||x||$. In the second one the general form of the characteristic functional of a normal measure in $l_p $ is found.

On Markov Random Sets

N. V. Krylov and A. A. Yushkevich

Theory Probab. Appl. 9, pp. 666-670 (5 pages) | Cited 2 times

Online Publication Date: July 28, 2006

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A Markov random set is a time-homogeneous random closed set on the half-line $t \geqq 0$, satisfying the Markov property of independence between the future and the past when the present is known. Such sets are introduced as a special class of Markov processes. They may be described by a non-increasing right-continuous positive function $g(x)$, $x > 0$, integrable near 0 and a non-negative number $\alpha $, determined up to an arbitrary positive constant factor. If $y(t)$ is a continuous strong Markov process, the $t$-set $\{ {y(t) = {\text{const}}} \}$ is a Markov random set. The most interesting Markov sets are obtained by simple transformations from the Brownien motion process.

A Nomogram Connecting the Parameters of Weibull’s Distribution with Probabilities

V. P. Kotel’nikov

Theory Probab. Appl. 9, pp. 670-674 (5 pages)

Online Publication Date: July 28, 2006

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The paper contains a compound nomogram designed for the determination of the parameter $m$ of Weibull’s distribution by statistical values of the mean $a_x $ and the standard deviation $\sigma _x $.
Besides the nomogram is useful for the determination of $P(X < x)$ for Weibull’s distribution by the parameters $m$, $a_x $ and $x$ or the quantity $x$ by $P(X < x)$, $m$ and $a_x $.

Uniformly Optimal Strategies in Search Problems

V. I. Arkin

Theory Probab. Appl. 9, pp. 674-680 (7 pages)

Online Publication Date: July 28, 2006

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Let $f(x)$ be the density function of the a priori distribution of a particle in $R^n $. The strategy of search is defined by a function $\alpha = \alpha (x,t) \geqq 0$, $\int_{R^n } {\alpha (x,t)} dx = 1$. The probability of finding the particle at a point $x$ during time $t$, under the condition that it is there, using the strategy $\alpha $, is given by the functional II $(\int_0^t {\alpha (x,t)dt,x} )$. Let $P_\alpha (T)$ be the probability of finding the particle using the strategy $\alpha $ during the time $T$. A strategy $\alpha ^ * $ is uniformly optimal if $P_{\alpha ^ * } (T) = \sup _\alpha P_\alpha (T)$ for any $T > 0$. In a very general case we prove the existence of the strategy $\alpha ^ * $ and find its explicit form.

Summary of Papers Presented at the Meetings of the Probability and Statistics Section of the Moscow Mathematical Society (March–May, 1964)

Theory Probab. Appl. 9, pp. 681-687 (7 pages)

Online Publication Date: July 28, 2006

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