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Theory of Probability and its Applications

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1960

Volume 5, Issue 4, pp. 341-431


Limit Theorems on the Distribution of Maximum of Sums of Bounded, Lattice Random Variables. II

A. A. Borovkov

Theory Probab. Appl. 5, pp. 341-355 (15 pages)

Online Publication Date: July 28, 2006

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The second part of this paper contains the proofs of integral theorems concerning \[ \bar s_n = \max \left( {\xi _1 + \xi _2 + \cdots + \xi _\nu } \right),\quad 1 \leqq \nu \leqq n \] where $\xi _i $ are bounded, lattice, independent and identically distributed random variables.

On the Theorems of Kolmogorov-Smirnov

D. A. Darling

Theory Probab. Appl. 5, pp. 356-361 (6 pages)

Online Publication Date: July 28, 2006

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The note contains, without proofs, exact and asymptotic representations of the distributions of the quantities $\sqrt{n}\sup |F_n (x) - F(x),\,N_n (a)$ (the number of zeros of the difference $F_n (x) - F(x) + a\sqrt n $ ), and of the Laplace transforms of the quantities $W_n^2 $ and $T_n (a)$ (the sum of the vertical parts of $F_n (x)$ which exceed $F(x) + a\sqrt n $.

Spectral Properties of Multivariate Stationary Processes and Boundary Properties of Analytic Matrices

Yu. A. Rozanov

Theory Probab. Appl. 5, pp. 362-376 (15 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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A survey is presented of some recent results obtained in the spectral theory of multivariate time series closely connected with the boundary properties of analytic matrices.

The Extreme Terms of a Sample and Their Role in the Sum of Independent Variables

D. Z. Arov and A. A. Bobrov

Theory Probab. Appl. 5, pp. 377-396 (20 pages) | Cited 6 times

Online Publication Date: July 28, 2006

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Let $(1)\qquad x_1 ,x_2 , \cdots ,x_n $ be independent random variables, whose distribution function $F(x)$ (the same for all $x_i $) satisfies the conditions of regularity of growth for $|x| \to \infty $: \[ \mathop {\lim }\limits_{x \to \infty } \frac{{\chi (kx)}}{{\chi (x)}} = \frac{1}{{k^\alpha }},\qquad 0 \leqq \alpha \leqq + \infty , \]\[ \mathop {\lim }\limits_{x \to \infty } \frac{{\chi ( - x)}}{{\chi (x)}} = c,\qquad 0 \leqq c \leqq + \infty , \] where \[ \chi (x) = 1 - F(x)\quad {\text{for}}\quad x > 0\quad {\text{and}}\quad chi (x) = F(x)\quad {\text{for}}\quad x < 0. \] If \[ (2)\qquad \xi _1^{(n)} ,\xi _2^{(n)} , \cdots ,\xi _{(n)}^{(n)} \]are the same random variables (1), which are written in descending order of their modules $|\xi _1^{(n)} | \geqq |\xi _2^{(n)} | \geqq \cdots \geqq |\xi _{(n)}^{(n)} |$ then $\xi _k^{(n)} $ for a given index $k < n$ is called the extreme term of (2), and the index $k$ is called its ordinal number.
The density of the joint limit distribution (for $n \to \infty $) of two properly normed extreme members $\xi _k^{(n)} $ and $\xi _m^{(n)} $ is established in § 1 and appropriate limit distributions for the ratio $\xi _k^{(n)} /\xi _m^{(n)} $ and other values are obtained. In § 2 we find the joint limit distribution of a properly normed extreme term $\xi _k^{(n)} $ and the sum \[ S_n^{(k)} = \left\{ \begin{gathered} \xi _{k + 1}^{(n)} + \cdots + \xi _n^{(n)} \quad {\text{for}}\quad 0 < \alpha < 1, \hfill \\ \xi _1^{(n)} + \cdots + \xi _n^{(n)} \quad {\text{for}}\quad 1 < \alpha < 2,a = \int_{ - \infty }^{ + \infty } {xdF(x)} . \hfill \\ \end{gathered} \right. \]
As a result, a limit distribution for $S_n^{(k)} /\xi _k^{(n)} $ is obtained. The method thus developed and the results obtained in §§ 1 and 2 are applied in § 3 to a more thorough study of the role of the extreme terms in the sum $S_n = x_1 + \cdots + x_n = \xi _1^{(n)} + \cdots + \xi _n^{(n)} $. Hence the total value of the first $k$ major terms $\xi _1^{(n)} , \cdots ,\xi _k^{(n)} $ in the sum Sn can be determined. The result obtained may be expressed as follows: \[ \begin{gathered} {\text{for}}\quad \alpha = 0,\quad S_n = \xi _1^{(n)} + \cdots + \xi _k^{(n)} + o(1)\xi _k^{(n)} , \hfill \\ \left. \begin{gathered} {\text{for}}\quad 0 < \alpha < 1,\quad S_n = \xi _1^{(n)} + \cdots + \xi _{k_n }^{(n)} + \left( {\frac{\alpha } {{1 - \alpha }} + o(1)} \right)k_n \xi _{k_n }^{(n)} \hfill \\ {\text{for}}\quad 1 < \alpha < 2,\quad S_n = na + o(1)k_n \xi _{k_n }^{(n)} \hfill \\ \end{gathered} \right\}\begin{array}{*{20}c} {k_n \to \infty ,} \\ {k_n = o\left( {\frac{n} {{\log n}}} \right),} \\ \end{array} \hfill \\ \end{gathered} \] where $o(1)$ tends in probability to 0 when $n \to \infty $.

On the Works of N. V. Smirnov In Mathematical Statistics

B. V. Gnedenko, A. N. Kolmogorov, Yu. V. Prokhorov, and O. V. Sarmanov

Theory Probab. Appl. 5, pp. 397-401 (5 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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Additive Functionals of a Wiener Process Determined by Stochastic Integrals

E. B. Dynkin

Theory Probab. Appl. 5, pp. 402-410 (9 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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We give definitions of additive and almost additive functionals of Markov processes and prove a general theorem about such functionals. Then we investigate the additive functionals of the $n$-dimensional Wiener process determined by stochastic integrals.

On Estimates of probabilities

L. N. Bol’shev

Theory Probab. Appl. 5, pp. 411-415 (5 pages) | Cited 10 times

Online Publication Date: July 28, 2006

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The problem of estimating an unknown probability in a series of independent trials is dealt -with. The experimental plan differs from the Bernoulli scheme. It is proved that the Clopper-Pearson method is applicable for the construction of interval estimates in an extensive class of :sequential type plans. The point and interval estimators for the Pólya scheme probability (negative binomial distribution) may serve as an example.
An approximate formula for confidence limits is given. This is an asymptotic formula of percentage points of a $B$-distribution. The precision of this formula is greater than the corresponding normal and Poisson approximations.

On Coalition-Free Strategies

N. N. Vorob’ev

Theory Probab. Appl. 5, pp. 415-417 (3 pages)

Online Publication Date: July 28, 2006

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A coalition-free strategy is a generalization of the concept of composite strategy proposed by Thompson. Any mixed strategy is equivalent to some coalition-free strategy.
Thus, generalizations are obtained of Kuhn’s theorem on the behavior of strategies and Thompson’s theorem on composite strategies.

Some Problems in the Spectral Theory of Higher-Order Moments, II

V. P. Leonov and A. N. Shiryaev

Theory Probab. Appl. 5, pp. 417-421 (5 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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This paper essentially contains a proof of the problem known in the technical literature as “The normalization of a wide-band stationary process when passing through a narrow-band filter” for processes of class $\Delta (\infty )$

An Approximation to the Distribution of Sums by Infinitely Divisible Laws

Yu. P. Studnev

Theory Probab. Appl. 5, pp. 421-425 (5 pages) | Cited 4 times

Online Publication Date: July 28, 2006

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The paper contains results on estimates of a “uniform distance” \[ \rho \left( {F^n ,\mathfrak{G}} \right) = \mathop {\inf }\limits_{S \in \mathfrak{G}} \mathop {\sup }\limits_x |F^n (x) - G(x)|, \] where $F^n (x) = \underbrace {F(x) * F(x) * \cdots * }_nF(x)$ and $\mathfrak{G}$ is the set of all infinitely divisible laws.

On the Limit Distribution of the First Jump

I. N. Kovalenko

Theory Probab. Appl. 5, pp. 425-428 (4 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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Formulas are obtained for the limit distribution of the first jump over the level $t$ when $t \to \infty $ for a sequence of identically distributed independent random variables with positive means. The cases of lattice as well as non-lattice distributions are discussed; the method of factorization [2] is used.

Polynomial Approximations and the Monte-Carlo Method

S. M. Ermakov and V. G. Zolotukhin

Theory Probab. Appl. 5, pp. 428-431 (4 pages) | Cited 5 times

Online Publication Date: July 28, 2006

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A new method of computing multiple integrals is proposed, which is a generalization of the ordinary Monte-Caro method.
This new method in evaluating the integral makes use of its approximate value as obtained by formulas for mechanical quadratures in accordance with a special distribution law for the integrational points.
This new method in evaluating the integral makes use of its approximate value as obtained by formulas for mechanical quadratures in accordance with a special distribution law for the integrational points.
It is shown that the standard deviation of the estimation may be considerably decreased, especially when the integrand possesses good differential properties.
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