Theory of Probability and Its Applications 

About: Theory of Probability and Its Applications is an academic journal published by Society for Industrial and Applied Mathematics. The journal publishes majorly in the area(s): Markov chain & Central limit theorem. It has an ISSN identifier of 0040-585X. Over the lifetime, 3672 publications have been published receiving 76665 citations.

On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities

Abstract: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady. The paper was first published in Russian as Вапник В. Н. and Червоненкис А. Я. О равномерноЙ сходимости частот появления событиЙ к их вероятностям. Теория вероятностеЙ и ее применения 16(2), 264–279 (1971).

On Estimating Regression

Abstract: A study is made of certain properties of an approximation to the regression line on the basis of sampling data when the sample size increases unboundedly.

Convergence of Random Processes and Limit Theorems in Probability Theory

Abstract: The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.).Chapter 1. Let $\Re $ be the c.s.m.s. and v a set of all finite measures on $\Re $. The distance $L(\mu _1 ,\mu _2 )$ (that is analogous to the Levy distance) is introduced, and equivalence of L-convergence and w. c. is proved. It is shown that $V\Re = (v,L)$ is c. s. m. s. Then, the necessary and sufficient conditions for compactness in $V\Re $ are given.In section 1.6 the concept of “characteristic functionals” is applied to the study of w. cc of measures in Hilbert space.Chapter 2. On the basis of the above results the necessary and sufficient compactness conditions for families of probability measures in spaces $C[0,1]$ and $D[0,1]$ (space of functions that are continuous in $[0,1]$ except for jumps) are formulated.Chapter 3. The general form of the “invariance principle” for the sums of independent random variables is developed....

On Optimum Methods in Quickest Detection Problems

Abstract: In this paper optimum methods are developed for observing a process (1), in which the moment when a “disorder” $\theta$ appears is not known. The basic quantity characterizing the quality of this observation method is the mean time delay ${\boldsymbol \tau}$ for detection of a disorder.After making assumption (4) it is shown that for a given false alarm probability $\omega$ or for a given ${\bf N}$ — mathematical expectation of false alarm numbers occurring up to the moment the disorder appears — the observation method minimizing ${\boldsymbol \tau } = {\boldsymbol \tau } (\omega)$ or ${\boldsymbol \tau} = {\boldsymbol \tau} ({\boldsymbol N})$ is based on an observation of a posteriors probability (23).In § 3 a case is considered, wherein, the disorder appears on the background of steadystate conditions arising when the disorder is absent. A method is found for minimizing ${\boldsymbol \tau} = {\boldsymbol \tau} ({\bf T})$ for a set ${\bf T}$ — mathematical expectation of the time between two false alarms...