## The boundedness of multiple series.(English. Ukrainian original)Zbl 0990.60048

Theory Probab. Math. Stat. 63, 99-108 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 90-99 (2000).
Let $$\{X(\overline n), \overline n\in N^{d}\}$$ be a field of random variables and let $$\{S(\overline n), \overline n\in N^{d}\}$$, $$S(\overline n)=\sum_{\overline k\leq\overline n}X(\overline k)$$, be a field of sums. The problem of boundedness in probability and almost sure boundedness of multiple series of independent variables is studied. The author proves that if $$\{X(\overline n), \overline n\in N^{d}\}$$ is a field of independent symmetric random variables, then convergence in probability of the field $$\{S(\overline n), \overline n\in N^{d}\}$$ is equivalent to its boundedness in probability. If $$\{X(\overline n), \overline n\in N^{d}\}$$ is a field of independent variables, then boundedness in probability of the field $$\{S(\overline n), \overline n\in N^{d}\}$$ is equivalent to almost sure boundedness. Necessary and sufficient conditions of almost sure boundedness of the field $$\{S(\overline n), \overline n\in N^{d}\}$$ are obtained. The multiple series $$\sum_{\overline n\in N^{d}} X(\overline n)$$ is called bounded convergent if it converges almost surely and the field $$\{S(\overline n), \overline n\in N^{d}\}$$ is bounded. The bounded convergence of multiple series is studied.

### MSC:

 60G50 Sums of independent random variables; random walks 60G60 Random fields