On the applicability conditions of the strong law of large numbers for sequences of independent random variables
It is well-known that Kolmogorov's condition \[ \sum^\infty_{n=1} {\operatorname{Var}X_n\over n^2}<\infty \] and Petrov's condition \[ \operatorname{Var}X_n= 0\Biggl({n^2\over \psi(n)}\Biggr),\tag{1} \] where \(\psi(n)\) is a positive nondecreasing function with the property \[ \sum^\infty_{n=1} {1\over n\psi(n)}< \infty,\tag{2} \] imply the strong low of large numbers for a sequence \(\{X_n\}_{n\geq 1}\) of independent random variables. The author studies the relationship between Kolmogorov's condition and Petrov's condition. The main result of the paper is expressed in the following theorem. Let \(\{b_n\}_{n\geq 1}\) be a sequence of positive numbers. Construct the function \(f(n)\), \(n= 0,1,\dots\), as follows \(f(0)= 1\), \(f(n)= n^2/\sum^n_{k=1} b_k\), for \(n\geq 1\). Conditions (1) and (2) imply \[ \sum^\infty_{n=1} {b_n\over n^2}< \infty.\tag{3} \] If condition (3) holds and there exists a nondecreasing function \(\psi(n)\asymp f(n)\) as \(n\to\infty\), then the function \(\psi\) satisfies conditions (1) and (2).
sequences of independent random variables
scientific article; zbMATH DE number 6109217