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Theory of Probability and its Applications

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1956

Volume 1, Issue 4, pp. 329-444


Central Limit Theorem for Nonstationary Markov Chains. II

R. L. Dobrushin

Theory Probab. Appl. 1, pp. 329-383 (55 pages) | Cited 5 times

Online Publication Date: July 28, 2006

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The second part of this paper contains proofs of results published in the first part (see the first number of this journal).

Two Uniform Limit Theorems for Sums of Independent Random Variables

A. N. Kolmogorov

Theory Probab. Appl. 1, pp. 384-394 (11 pages) | Cited 21 times

Online Publication Date: July 28, 2006

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Let $\xi _1 , \cdots ,\xi _n $ be independent random variables,\[ \xi = \xi _1 , \cdots ,\xi _n , \]and $\Phi ,F_1 , \cdots ,F_n $ the corresponding distribution functions. Let us denote by $\mathfrak{E}$ the totality of degenerate distributions\[ E(x)\left\{ \begin{gathered} 0{\text{ for }}x \leqq a \hfill \\ 1{\text{ for }}x > a \hfill \\ \end{gathered} \right. \]and by $\Theta $ the totality of infinitely divisible distributions.
Theorem 1.There exists a constant$C$such that for all$\varepsilon > 0,L > 2l > 0$the inequalities\[ E_k (x - l) - \varepsilon \leqq F_k (x) \leqq E_k (x + l) + \varepsilon ,E_k \in \mathfrak{E},\quad k = 1, \cdots ,n; - \infty < x < + \infty , \]imply the existence of a function$\Psi \in \Theta $such that\[ \Psi (x - L) - \delta \leqq \Phi (x) \leqq \Psi (x + L) + \delta ,\quad - \infty < x < + \infty , \]where\[ \delta = C\max \left( {\frac{L} {l}\sqrt {\log \frac{L} {l}} ,\varepsilon ^{{1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-\nulldelimiterspace} 5}} } \right). \]
In the case of identical distributions $F_k $,\[ F_k (x) \equiv F(x),\quad k = 1, \cdots ,n,\]a simpler theorem may be proved:
Theorem 2. There exists a constant$C$such that for all$n$and$F$one can find a function$\Psi \in \Theta $satisfying the inequalities\[ \left| \Psi (x) - \Theta (x)\right| \leqq Cn^{{1 / 5}} , - \infty < x < + \infty .\]

Sequential Bayes Solutions and Optimal Methods of Statistical Acceptance Control

V. S. Mikhalevish

Theory Probab. Appl. 1, pp. 395-421 (27 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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This paper gives a description of classes of sequential Bares solutions for the binomial family of distributions and for the Poisson process, if the weight function is monotonic and some of its zeros are points of increase for the a priori probability distribution. The corresponding Bares solutions are truncated with respect to $n$ (or $t$). These solutions are determined by the process of trial and error between two bounds. A risk function for a Poisson process is proved to satisfy certain differential equations. It is possible to determine the bounds of Bares solutions with the help of these equations.
The limiting theorem on the convergence of the boundaries of Bares solutions and the risk functions, when the corresponding binomial processes converge to a Poisson process, is proved. The results obtained are used in describing the class of optimum methods of statistical acceptance control.

On the Question of Finding a General Distribution from the Distribution of a Statistic

Yu. V. Linnik

Theory Probab. Appl. 1, pp. 422-434 (13 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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Let $X$ be a real random variable with the distribution function $F(x) = {\bf P}(X < x)$ and $\xi = (x_1 , \cdots ,x_n )$ the corresponding sample of size $n$ ($x_i $ being independent replicas of $X$).
A statistic $Q(\xi )$ is called definite if it is homogeneous of positive dimension and the level surfaces $Q(\xi ) = {\text{const.}}$ are continuous, piecewise-smooth and star-finite regions. A statistic $Q(\xi )$ is called defining in a class $K$ of distribution functions $F(x)$, if the distribution $F_Q (x) = {\bf P}(Q < x)$, induced by $F(x)$, determines $F(x)$ in the class $K$. A definite statistic cannot be defining in general for the class $K$ of all distribution functions, but it is defining in certain rather wide classes of symmetric distribution densities. Three theorems are proved to this effect. The problem can be given as a generalization of the classical moment problem, putting $Q(\xi ) = x_1^2 + \cdots + x_n^2 $.

A Remark on Cramer’s Theorem on the Decomposition of the Normal Law

Yu. V. Linnik

Theory Probab. Appl. 1, pp. 435-436 (2 pages) | Cited 2 times

Online Publication Date: July 28, 2006

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H. Cramer’s theorem is shown to be an equivalent of a particular case of Skitovich-Darmois’ theorem on the independence of linear statistics. This means that each one of these theorems can be deduced from the other by a short elementary argument.

An Example of a Countable Homogeneous Markov Process All States of which are Transient

R. L Dobrushin

Theory Probab. Appl. 1, pp. 436-440 (5 pages)

Online Publication Date: July 28, 2006

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An example of a homogeneous Markov process is composed with a countable set of states, all densities of outcome probabilities of which are infinite.

A Theorem in the Theory of Infinitely Divisible Laws

I. A. Ibragimov

Theory Probab. Appl. 1, pp. 440-444 (5 pages)

Online Publication Date: July 28, 2006

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Let $\mathfrak{F}$ be a class of infinitely divisible distribution functions $F$ for which, if $F \in \mathfrak{F}$ and $F * H = Q$, where $Q(x)$ is an infinitely divisible distribution function, it follows that $H$ is also an infinitely divisible distribution function. The following theorem is proved:
Class$\mathfrak{F}$is identical to the set o f all normal distributions.
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