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Theory of Probability and its Applications

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1957

Volume 2, Issue 4, pp. 378-480


Some Limit Theorems for Stationary Markov Chains

S. V. Nagaev

Theory Probab. Appl. 2, pp. 378-406 (29 pages) | Cited 28 times

Online Publication Date: July 17, 2006

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Let $X$ be a space of points, $F_X $ a $\sigma $-algebra of its subsets, and $p(\xi ,A)$, $\xi \in X$, $A \in F_X $, a stochastic transition function satisfying the following condition:
an integer $k \geqq 1$ exists such that \[ {\text{(1)}}\qquad \mathop {\sup }\limits_{\eta ,\xi \in X,A \in F_X } \left| {p^{(k)} (\xi ,A) - p^{(k)} (\eta ,A)} \right| < 1. \] Let us define the sequence of random variables $x_1 ,x_2 , \cdots ,x_n , \cdots$ as follows: \[ \Pr \left( {x_1 \in A_1 ,x_2 \in A_2 , \cdots ,x_n \in A_n } \right) = \int\limits_{A_1 } {\pi (d\xi _1 )} \int\limits_{A_2 } {p(\xi _1 ,d\xi _2 ) \cdots } \int\limits_{A_n } {p(\xi _{n - 1} ,d\xi _n )} , \] where $\pi ( \cdot )$ is the initial distribution.
Let $f(\xi )$ be a real function of $\xi \in X$ measurable with respect to $F_X $.
In Chapter I the asymptotic behaviour of the characteristic function of $\sum _1^n f(x_i )$ is studied. Chapter II is devoted to limit theorems. The central limit theorem is proved under the assumption that \[ {\text{(2)}}\qquad \int\limits_X {f^2 (\xi )p(d\xi ) < \infty } , \] where $p( \cdot )$ is a stationary absolute probability distribution corresponding to $p( \cdot , \cdot )$. The sufficient conditions for convergence to stable laws are given. In chapter III the local limit theorem is proved, and asumptotic expansions are given. The characteristic function method is the basic one used.

On the Differentiability of Measures Which Correspond to Stochastic Processes. I. Processes with Independent Increments

A. V. Skorokhod

Theory Probab. Appl. 2, pp. 407-432 (26 pages) | Cited 5 times

Online Publication Date: July 17, 2006

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Kolmogorov (see [2] pg. 39) has proved that for each stochastic process there exists a corresponding unique measure on the minimal Borel field containing all cylindrical sets of the space of all functions.
Let $\xi _1 (t)$ and $\xi _2 (t)$ be processes with independent increments and $\mu _1 $ and $\mu _2 $ — measures corresponding to these processes. In this paper the conditions for which the measure $\mu _2 $ is absolutely continuous with respect to the measure $\mu _1 $ are investigated (Theorem A), and the density of the measure $\mu _2 $ with respect to the measure $\mu _2 $ is calculated (Theorem B).

Mellin-Stieltjes Transforms in Probability Theory

V. M. Zolotarev

Theory Probab. Appl. 2, pp. 433-460 (28 pages) | Cited 12 times

Online Publication Date: July 17, 2006

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Mellin-Stieltjes transforms are very useful in solving problems in which products and ratios of random variables are encountered. The paper relates some general considerations pertaining to the application of these transforms (Section 1), and also gives a concrete example of their use in studying analytical properties of stable distributions (Section 2). In the first section the relationship between the Mellin-Stieltjes transform, the unilateral Laplace-Stieltjes transform and the characteristic function of a given distribution is established. For the sake of simplicity, all distributions in Section 1 are considered as being continuous at zero. The concepts “truncation” and equivalence of random variables are also introduced there.
Any random variable having a distribution function \[ \tilde F(x) = \frac{{F(x) - F(0)}}{{1 - F(0)}},\ x \geqq 0;\quad \tilde F(x) \equiv 0,\ x \leqq 0 \] is called the truncation$\xi $of the random variable$\xi $having a distribution function$F(x)$.
Truncations may also-be considered as functions of an initial random variable $\xi $ (two different representations of $\xi $ are given as a function of $\xi $). Random variables $\xi _1 $ and $\xi _2 $ are considered as being equivalent and are designated as $\xi _1 \approx \xi _2 $ if the distribution functions corresponding to them are equal.
The concepts of truncations and equivalency of random variables are systematically employed in the second section in establishing precise and limiting relationships in the class of stable distributions. Stable distributions naturally decompose into two analytically independent branches if the shift parameter $\gamma $ is specially selected. For $\alpha \ne 1$ it is possible to determine an explicit expression of the Mellin transform for these branches employing Euler’s $\Gamma $-function. These two circumstances justify the use of the above mentioned concepts.
This explicit representation of the Mellin transforms in turn is used to determine a whole series of relationships between branches of stable distributions, in which all previously known relationships of the same type are special cases.
It should be noted that in paragraph 2.6 all random variables, which are written separately, are considered independent. The behaviour of stable distributions near critical points $\alpha = 0$ and $\alpha = 1$ is investigated in the second section.

A Nomogram for the Incomplete $\Gamma $-Function and the $\chi ^2 $ Probability Function

S. V. Smirnov and M. K. Potapov

Theory Probab. Appl. 2, pp. 461-465 (5 pages)

Online Publication Date: July 17, 2006

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A nomogram is constructed of the function $P(\chi ^2 ,n) = 1 - \Gamma (m,y)$. For $n \geqq 30$ the function $\Pi $ is introduced, which is obtained from $P$ by means of the transformation $t = \sqrt {2\chi ^2 } - \sqrt {2n} ,x = \sqrt {2/n} $, while for $1 \leqq n \leqq 30$ the function $P$ itself is considered.
The nomogram is valid for the following values of $n,t,\chi ^2 $ and $P:1 \leqq n \leqq \infty $; $| t | \leqq 3.1$; $1 \leqq \chi ^2 \leqq 30;0.001 \leqq P \leqq 0.999$; $0.001 \leqq P \leqq 0.999$. The absolute error in the entire nomogram for $0.01 \leqq P \leqq 0.99$ is found not to exceed $0.005$.

Non-Linear Confluence Analysis

N. P. Klepikov and S. N. Sokolov

Theory Probab. Appl. 2, pp. 465-468 (4 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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Non-linear confluence analysis, necessary for the treatment of experimental data when all variables are subject to errors, is considered from the standpoint of the maximum likelihood method. The likelihood function is a product of curvilinear integrals of the respective distribution densities of each point of the curve. For a sufficiently small curvature and a normal error distribution, these integrals are evaluated approximately, resulting in distribution functions of the normal type but with modified weights and shifted experimental points. Thus, a confluent problem is reduced to an ordinary regressional one. Weight modifications and point shifts may be found by means of successive approximations.

A Characterization of Normal Distributions in Hilbert Space

Yu. V. Prokhorov and M. Fish

Theory Probab. Appl. 2, pp. 468-470 (3 pages)

Online Publication Date: July 17, 2006

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Theorem 1 gives an extension of the well known result of [2] to the case of random elements in Hilbert space.

Summary of Papers Presented at the Sessions of the Scientific Research Seminar on Probability Theory, (Moscow, February–May, 1957)

Theory Probab. Appl. 2, pp. 470-480 (11 pages)

Online Publication Date: July 17, 2006

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