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

Klesov, O. I.

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.

Reviewer: A.D.Borisenko (Kyïv)

MSC:

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

Keywords:

multiple series; boundedness in probability; almost sure boundedness; random field; independent random variables