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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1960, Volume 5, Issue 1, Pages 54–83 (Mi tvp4814)

On Some Limit Theorems of Probability Theory

K. V. Maslov

Kharkov

Abstract: The following problem is considered in the paper. Let
$$\xi_{n,1},\xi_{n,2},…,\xi_{n,k_n},\quad n=1,2,…,$$
be a sequence of a series of independent random variables; $\varphi (x,y)$ is any function of two variables and the random variables $\zeta_{n,k}$ are determined as
$$\zeta_{n,1}=\xi_{n,1},\zeta_{n,k+1}=\varphi({\zeta_{n,k},\xi_{n,k+1}}),\quad k=1,2,…k_n-1.$$

We look for sufficient conditions for the existence of a limit distribution of random variable $\zeta_{n,k_n},n\to\infty$, and the form of this distribution. If $\varphi (x,y)=x+y$ we have the well-known problems for sums of independent random variables.
Our method of solution of the formulated problems is different from the methods usually employed in analogous studies (e.g. from S. N. Bernstein's methods, which were developed in [2] for solution to a similar problem).
The theory of partial differential equations and the theory of Markov processes are our basic tools.

Full text: PDF file (2751 kB)

English version:
Theory of Probability and its Applications, 1960, 5:1, 50–79

Citation: K. V. Maslov, “On Some Limit Theorems of Probability Theory”, Teor. Veroyatnost. i Primenen., 5:1 (1960), 54–83; Theory Probab. Appl., 5:1 (1960), 50–79

Citation in format AMSBIB
\Bibitem{Mas60} \by K.~V.~Maslov \paper On Some Limit Theorems of Probability Theory \jour Teor. Veroyatnost. i Primenen. \yr 1960 \vol 5 \issue 1 \pages 54--83 \mathnet{http://mi.mathnet.ru/tvp4814} \transl \jour Theory Probab. Appl. \yr 1960 \vol 5 \issue 1 \pages 50--79 \crossref{https://doi.org/10.1137/1105006}