On Some Limit Theorems of Probability Theory

K. V. Maslov

Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 1, pp. 54-83

Résumé

The following problem is considered in the paper. Let $$\xi_{n,1},\xi_{n,2},\dots,\xi_{n,k_n},\quad n=1,2,\dots,$$ 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 $$\zeta_{n,1}=\xi_{n,1},\zeta_{n,k+1}=\varphi\left({\zeta_{n,k},\xi_{n,k+1}}\right),\quad k=1,2,\dots k_n-1.$$ 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.