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

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