The law of large numbers for multiple sums of independent identically distributed random variables. (English. Russian original) Zbl 0858.60021

Klesov, O. I.

Theory Probab. Math. Stat. 50, 77-87 (1995); translation from Teor. Jmovirn. Mat. Stat. 50, 76-86 (1994).

In research on the law of large numbers for independent identically distributed random variables indexed by \(d\)-dimensional indices \(\overline{n}=(n_1,\dots,n_d)\), \(d \geq 1\), various definitions of convergence have been used. Convergences considered are the following: \(\min(n_1,\dots,n_d) \to \infty\) and \(\max(n_1,\dots,n_d) \to \infty\). The author presents necessary and sufficient conditions under which the weak law of large numbers is satisfied. Let \(\{X_j(\overline{n}), j \leq j(\overline{n})\}\) be a finite collection of independent random variables, where \(j(\overline{n})\) is an integer number. Let \(S(\overline{n}) = \sum_{j\leq j(\overline{n})} X_j(\overline{n})\). Consider the following conditions: (1) \(\lim P (|S(\overline{n})|\geq t)=0\) for all \(t>0\), (2) for any \(\varepsilon > 0\) \(\lim\max_{j\leq j(\overline{n})}P(|X_j(\overline{n})|\geq \varepsilon)=0\), (3) \(\lim \sum_{j\leq j(\overline{n})} P(|X_j(\overline{n})|\geq \varepsilon)=0\), for some \(\varepsilon >0\), (4) \(\lim \sum_{j\leq j(\overline{n})} EX^c_j(\overline{n})=0\), for some \(c>0\), and (5) \(\lim\sum_{j\leq j(\overline{n})} \text{Var }X^c_j(\overline{n})=0\), for some \(c>0\). The main result: conditions (1) and (2) are equivalent to conditions (3), (4), and (5).

Reviewer: T.I.Jeon (Taejon)

MSC:

60F05	Central limit and other weak theorems
60G50	Sums of independent random variables; random walks

Keywords:

law of large numbers; various definitions of convergence